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is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Zhang%2C+A&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Zhang%2C+A&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Zhang%2C+A&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Zhang%2C+A&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.11355">arXiv:2503.11355</a> <span> [<a href="https://arxiv.org/pdf/2503.11355">pdf</a>, <a href="https://arxiv.org/ps/2503.11355">ps</a>, <a href="https://arxiv.org/format/2503.11355">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Software">cs.MS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Software Engineering">cs.SE</span> </div> </div> <p class="title is-5 mathjax"> TypedMatrices.jl: An Extensible and Type-Based Matrix Collection for Julia </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A">Anzhi Zhang</a>, <a href="/search/math?searchtype=author&query=Fasi%2C+M">Massimiliano Fasi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.11355v1-abstract-short" style="display: inline;"> TypedMatrices.jl is a Julia package to organize test matrices. By default, the package comes with a number of built-in matrices and interfaces to help users select test cases based on their properties. The package is designed to be extensible, allowing users to define their own matrix types. We discuss the design and implementation of the package and demonstrate its usage with a number of examples… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.11355v1-abstract-full').style.display = 'inline'; document.getElementById('2503.11355v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.11355v1-abstract-full" style="display: none;"> TypedMatrices.jl is a Julia package to organize test matrices. By default, the package comes with a number of built-in matrices and interfaces to help users select test cases based on their properties. The package is designed to be extensible, allowing users to define their own matrix types. We discuss the design and implementation of the package and demonstrate its usage with a number of examples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.11355v1-abstract-full').style.display = 'none'; document.getElementById('2503.11355v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.08504">arXiv:2503.08504</a> <span> [<a href="https://arxiv.org/pdf/2503.08504">pdf</a>, <a href="https://arxiv.org/format/2503.08504">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> Strichartz estimates for orthonormal systems on compact manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+X">Xing Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">An Zhang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+C">Cheng Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.08504v1-abstract-short" style="display: inline;"> We establish new Strichartz estimates for orthonormal systems on compact Riemannian manifolds in the case of wave, Klein-Gordon and fractional Schr枚dinger equations. Our results generalize the classical (single-function) Strichartz estimates on compact manifolds by Kapitanski, Burq-G茅rard-Tzvetkov, Dinh, and extend the Euclidean orthonormal version by Frank-Lewin-Lieb-Seiringer, Frank-Sabin, Bez-L… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.08504v1-abstract-full').style.display = 'inline'; document.getElementById('2503.08504v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.08504v1-abstract-full" style="display: none;"> We establish new Strichartz estimates for orthonormal systems on compact Riemannian manifolds in the case of wave, Klein-Gordon and fractional Schr枚dinger equations. Our results generalize the classical (single-function) Strichartz estimates on compact manifolds by Kapitanski, Burq-G茅rard-Tzvetkov, Dinh, and extend the Euclidean orthonormal version by Frank-Lewin-Lieb-Seiringer, Frank-Sabin, Bez-Lee-Nakamura. On the flat torus, our new results cover prior work of Nakamura for the Schr枚dinger equation, which exploits the dispersive estimate of Kenig-Ponce-Vega. We achieve sharp results on compact manifolds by combining the frequency localized dispersive estimates for small time intervals with the duality principle due to Frank-Sabin. We observe a new phenomenon that the results in the supercritical regime are sensitive to the geometry of the manifold. Moreover, we establish sharp Strichartz estimates on the flat torus for the fractional Schr枚dinger equations by proving a new decoupling inequality for certain non-smooth hypersurfaces. As applications, we obtain the well-posedness of infinite systems of dispersive equations with Hartree-type nonlinearity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.08504v1-abstract-full').style.display = 'none'; document.getElementById('2503.08504v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58J45; 58J50; 58J40; 35B45 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.06715">arXiv:2503.06715</a> <span> [<a href="https://arxiv.org/pdf/2503.06715">pdf</a>, <a href="https://arxiv.org/ps/2503.06715">ps</a>, <a href="https://arxiv.org/format/2503.06715">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Morita Equivalence of Subrings with Applications to Inverse Semigroup Algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A">Allen Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.06715v1-abstract-short" style="display: inline;"> We develop a technique to show the Morita equivalence of certain subrings of a ring with local units. We then apply this technique to develop conditions that are sufficient to show the Morita equivalence of subalgebras induced by partial subactions on generalized Boolean algebras and, subsequently, strongly $E^{\ast}$-unitary inverse subsemigroups. As an application, we prove that the Leavitt path… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.06715v1-abstract-full').style.display = 'inline'; document.getElementById('2503.06715v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.06715v1-abstract-full" style="display: none;"> We develop a technique to show the Morita equivalence of certain subrings of a ring with local units. We then apply this technique to develop conditions that are sufficient to show the Morita equivalence of subalgebras induced by partial subactions on generalized Boolean algebras and, subsequently, strongly $E^{\ast}$-unitary inverse subsemigroups. As an application, we prove that the Leavitt path algebra of a graph is Morita equivalent to the Leavitt path algebra of certain subgraphs and use this to calculate the Morita equivalence class of some Leavitt path algebras. Finally, as the main application, we prove a desingularization result for labelled Leavitt path algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.06715v1-abstract-full').style.display = 'none'; document.getElementById('2503.06715v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 16D90 (Primary) 16S35; 20M18; 16S88 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.01644">arXiv:2503.01644</a> <span> [<a href="https://arxiv.org/pdf/2503.01644">pdf</a>, <a href="https://arxiv.org/ps/2503.01644">ps</a>, <a href="https://arxiv.org/format/2503.01644">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Partial Actions on Generalized Boolean Algebras with Applications to Inverse Semigroups and Combinatorial $R$-Algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A">Allen Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.01644v1-abstract-short" style="display: inline;"> We define the notion of a partial action on a generalized Boolean algebra and associate to every such system and commutative unital ring $R$ an $R$-algebra. We prove that every strongly $E^{\ast}$-unitary inverse semigroup has an associated partial action on the generalized Boolean algebra of compact open sets of tight filters in the meet semilattice of idempotents. Using these correspondences, we… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.01644v1-abstract-full').style.display = 'inline'; document.getElementById('2503.01644v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.01644v1-abstract-full" style="display: none;"> We define the notion of a partial action on a generalized Boolean algebra and associate to every such system and commutative unital ring $R$ an $R$-algebra. We prove that every strongly $E^{\ast}$-unitary inverse semigroup has an associated partial action on the generalized Boolean algebra of compact open sets of tight filters in the meet semilattice of idempotents. Using these correspondences, we associate to every strongly $E^{\ast}$-unitary inverse semigroup and commutative unital ring $R$ an $R$-algebra, and show that it is isomorphic to the Steinberg algebra of the tight groupoid. As an application, we show that there is a natural unitization operation on an inverse semigroup that corresponds to a unitization of the corresponding $R$-algebra. Finally, we show that Leavitt path algebras and labelled Leavitt path algebras can be realized as the $R$-algebra associated to a strongly $E^{\ast}$-unitary inverse semigroup. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.01644v1-abstract-full').style.display = 'none'; document.getElementById('2503.01644v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20M30 (Primary) 16S35; 16S88 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.09946">arXiv:2501.09946</a> <span> [<a href="https://arxiv.org/pdf/2501.09946">pdf</a>, <a href="https://arxiv.org/ps/2501.09946">ps</a>, <a href="https://arxiv.org/format/2501.09946">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Client-Centric Federated Adaptive Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Sun%2C+J">Jianhui Sun</a>, <a href="/search/math?searchtype=author&query=Wu%2C+X">Xidong Wu</a>, <a href="/search/math?searchtype=author&query=Huang%2C+H">Heng Huang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Aidong Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.09946v1-abstract-short" style="display: inline;"> Federated Learning (FL) is a distributed learning paradigm where clients collaboratively train a model while keeping their own data private. With an increasing scale of clients and models, FL encounters two key challenges, client drift due to a high degree of statistical/system heterogeneity, and lack of adaptivity. However, most existing FL research is based on unrealistic assumptions that virtua… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.09946v1-abstract-full').style.display = 'inline'; document.getElementById('2501.09946v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.09946v1-abstract-full" style="display: none;"> Federated Learning (FL) is a distributed learning paradigm where clients collaboratively train a model while keeping their own data private. With an increasing scale of clients and models, FL encounters two key challenges, client drift due to a high degree of statistical/system heterogeneity, and lack of adaptivity. However, most existing FL research is based on unrealistic assumptions that virtually ignore system heterogeneity. In this paper, we propose Client-Centric Federated Adaptive Optimization, which is a class of novel federated adaptive optimization approaches. We enable several features in this framework such as arbitrary client participation, asynchronous server aggregation, and heterogeneous local computing, which are ubiquitous in real-world FL systems but are missed in most existing works. We provide a rigorous convergence analysis of our proposed framework for general nonconvex objectives, which is shown to converge with the best-known rate. Extensive experiments show that our approaches consistently outperform the baseline by a large margin across benchmarks. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.09946v1-abstract-full').style.display = 'none'; document.getElementById('2501.09946v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.06652">arXiv:2501.06652</a> <span> [<a href="https://arxiv.org/pdf/2501.06652">pdf</a>, <a href="https://arxiv.org/format/2501.06652">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> High-order Accurate Inference on Manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+C">Chengzhu Huang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.06652v2-abstract-short" style="display: inline;"> We present a new framework for statistical inference on Riemannian manifolds that achieves high-order accuracy, addressing the challenges posed by non-Euclidean parameter spaces frequently encountered in modern data science. Our approach leverages a novel and computationally efficient procedure to reach higher-order asymptotic precision. In particular, we develop a bootstrap algorithm on Riemannia… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.06652v2-abstract-full').style.display = 'inline'; document.getElementById('2501.06652v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.06652v2-abstract-full" style="display: none;"> We present a new framework for statistical inference on Riemannian manifolds that achieves high-order accuracy, addressing the challenges posed by non-Euclidean parameter spaces frequently encountered in modern data science. Our approach leverages a novel and computationally efficient procedure to reach higher-order asymptotic precision. In particular, we develop a bootstrap algorithm on Riemannian manifolds that is both computationally efficient and accurate for hypothesis testing and confidence region construction. Although locational hypothesis testing can be reformulated as a standard Euclidean problem, constructing high-order accurate confidence regions necessitates careful treatment of manifold geometry. To this end, we establish high-order asymptotics under a fixed normal chart centered at the true parameter, thereby enabling precise expansions that incorporate curvature effects. We demonstrate the versatility of this framework across various manifold settings-including spheres, the Stiefel manifold, fixed-rank matrices manifolds, and rank-one tensor manifolds-and, for Euclidean submanifolds, introduce a class of projection-like coordinate charts with strong consistency properties. Finally, numerical studies confirm the practical merits of the proposed procedure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.06652v2-abstract-full').style.display = 'none'; document.getElementById('2501.06652v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.19049">arXiv:2411.19049</a> <span> [<a href="https://arxiv.org/pdf/2411.19049">pdf</a>, <a href="https://arxiv.org/format/2411.19049">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> A dichotomy theorem on the complexity of 3-uniform hypergraphic degree sequence graphicality </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Logsdon%2C+S">Sara Logsdon</a>, <a href="/search/math?searchtype=author&query=Maheshwari%2C+A">Arya Maheshwari</a>, <a href="/search/math?searchtype=author&query=Mikl%C3%B3s%2C+I">Istv谩n Mikl贸s</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Angelina Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.19049v1-abstract-short" style="display: inline;"> We present a dichotomy theorem on the parameterized complexity of the 3-uniform hypergraphicality problem. Given $0<c_1\le c_2 < 1$, the parameterized 3-uniform Hypergraphic Degree Sequence problem, $3uni-HDS_{c_1,c_2}$, considers degree sequences $D$ of length $n$ such that all degrees are between $c_1 {n-1 \choose 2}$ and $c_2 {n-1\choose 2}$ and it asks if there is a 3-uniform hypergraph with d… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.19049v1-abstract-full').style.display = 'inline'; document.getElementById('2411.19049v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.19049v1-abstract-full" style="display: none;"> We present a dichotomy theorem on the parameterized complexity of the 3-uniform hypergraphicality problem. Given $0<c_1\le c_2 < 1$, the parameterized 3-uniform Hypergraphic Degree Sequence problem, $3uni-HDS_{c_1,c_2}$, considers degree sequences $D$ of length $n$ such that all degrees are between $c_1 {n-1 \choose 2}$ and $c_2 {n-1\choose 2}$ and it asks if there is a 3-uniform hypergraph with degree sequence $D$. We prove that for any $0<c_2< 1$, there exists a unique, polynomial-time computable $c_1^*$ with the following properties. For any $ c_1\in (c_1^*,c_2]$, $3uni-HDS_{c_1,c_2}$ can be solved in linear time. In fact, for any $c_1\in (c_1^*,c_2]$ there exists an easy-to-compute $n_0$ such that any degree sequence $D$ of length $n\ge n_0$ and all degrees between $c_1 {n-1\choose 2}$ and $c_2 {n-1\choose 2}$ has a 3-uniform hypergraph realization if and only if the sum of the degrees can be divided by $3$. Further, $n_0$ grows polynomially with the inverse of $c_1-c_1^*$. On the other hand, we prove that for all $c_1<c_1^*$, $3uni-HDS_{c_1,c_2}$ is NP-complete. Finally, we briefly consider an extension of the hypergraphicality problem to arbitrary $t$-uniformity. We show that the interval where degree sequences (satisfying divisibility conditions) always have $t$-uniform hypergraph realizations must become increasingly narrow, with interval width tending to $0$ as $t \rightarrow \infty$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.19049v1-abstract-full').style.display = 'none'; document.getElementById('2411.19049v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> MSC 2020 05C65; 05C07; 05C85; 68Q17; 68Q27 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.17151">arXiv:2411.17151</a> <span> [<a href="https://arxiv.org/pdf/2411.17151">pdf</a>, <a href="https://arxiv.org/ps/2411.17151">ps</a>, <a href="https://arxiv.org/format/2411.17151">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Random attractors for damped stochastic fractional Schr枚dinger equation on $\mathbb{R}^{n}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+L">Li Lin</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yanjie Zhang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Ao Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.17151v2-abstract-short" style="display: inline;"> We study the random attractors associated with the stochastic fractional Schr枚dinger equation on $\mathbb{R}^n$. Utilizing the stochastic Strichartz estimates for the damped fractional Schr枚dinger equation with Gaussian noise, we show the existence and uniqueness of a global solution to the damped stochastic fractional nonlinear Schr枚dinger equation in $H^伪(\mathbb{R}^n)$. Furthermore, we demonstr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.17151v2-abstract-full').style.display = 'inline'; document.getElementById('2411.17151v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.17151v2-abstract-full" style="display: none;"> We study the random attractors associated with the stochastic fractional Schr枚dinger equation on $\mathbb{R}^n$. Utilizing the stochastic Strichartz estimates for the damped fractional Schr枚dinger equation with Gaussian noise, we show the existence and uniqueness of a global solution to the damped stochastic fractional nonlinear Schr枚dinger equation in $H^伪(\mathbb{R}^n)$. Furthermore, we demonstrate that this equation defines an infinite-dimensional dynamical system, which possesses a global attractor in $H^伪(\mathbb{R}^n)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.17151v2-abstract-full').style.display = 'none'; document.getElementById('2411.17151v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: text overlap with arXiv:2411.02781</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.15660">arXiv:2411.15660</a> <span> [<a href="https://arxiv.org/pdf/2411.15660">pdf</a>, <a href="https://arxiv.org/format/2411.15660">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Federated PCA and Estimation for Spiked Covariance Matrices: Optimal Rates and Efficient Algorithm </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+J">Jingyang Li</a>, <a href="/search/math?searchtype=author&query=Cai%2C+T+T">T. Tony Cai</a>, <a href="/search/math?searchtype=author&query=Xia%2C+D">Dong Xia</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.15660v1-abstract-short" style="display: inline;"> Federated Learning (FL) has gained significant recent attention in machine learning for its enhanced privacy and data security, making it indispensable in fields such as healthcare, finance, and personalized services. This paper investigates federated PCA and estimation for spiked covariance matrices under distributed differential privacy constraints. We establish minimax rates of convergence, wit… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.15660v1-abstract-full').style.display = 'inline'; document.getElementById('2411.15660v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.15660v1-abstract-full" style="display: none;"> Federated Learning (FL) has gained significant recent attention in machine learning for its enhanced privacy and data security, making it indispensable in fields such as healthcare, finance, and personalized services. This paper investigates federated PCA and estimation for spiked covariance matrices under distributed differential privacy constraints. We establish minimax rates of convergence, with a key finding that the central server's optimal rate is the harmonic mean of the local clients' minimax rates. This guarantees consistent estimation at the central server as long as at least one local client provides consistent results. Notably, consistency is maintained even if some local estimators are inconsistent, provided there are enough clients. These findings highlight the robustness and scalability of FL for reliable statistical inference under privacy constraints. To establish minimax lower bounds, we derive a matrix version of van Trees' inequality, which is of independent interest. Furthermore, we propose an efficient algorithm that preserves differential privacy while achieving near-optimal rates at the central server, up to a logarithmic factor. We address significant technical challenges in analyzing this algorithm, which involves a three-layer spectral decomposition. Numerical performance of the proposed algorithm is investigated using both simulated and real data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.15660v1-abstract-full').style.display = 'none'; document.getElementById('2411.15660v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.02781">arXiv:2411.02781</a> <span> [<a href="https://arxiv.org/pdf/2411.02781">pdf</a>, <a href="https://arxiv.org/ps/2411.02781">ps</a>, <a href="https://arxiv.org/format/2411.02781">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Weak pullback attractors for damped stochastic fractional Schr枚dinger equation on $\mathbb{R}^n </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A">Ao Zhang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yanjie Zhang</a>, <a href="/search/math?searchtype=author&query=Zhai%2C+S">Sanyang Zhai</a>, <a href="/search/math?searchtype=author&query=Lin%2C+L">Li Lin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.02781v1-abstract-short" style="display: inline;"> This article discusses the weak pullback attractors for a damped stochastic fractional Schr枚dinger equation on $\mathbb{R}^n$ with $n\geq 2$. By utilizing the stochastic Strichartz estimates and a stopping time technique argument, the existence and uniqueness of a global solution for the systems with the nonlinear term $|u|^{2蟽}u$ are proven. Furthermore, we define a mean random dynamical system d… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.02781v1-abstract-full').style.display = 'inline'; document.getElementById('2411.02781v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.02781v1-abstract-full" style="display: none;"> This article discusses the weak pullback attractors for a damped stochastic fractional Schr枚dinger equation on $\mathbb{R}^n$ with $n\geq 2$. By utilizing the stochastic Strichartz estimates and a stopping time technique argument, the existence and uniqueness of a global solution for the systems with the nonlinear term $|u|^{2蟽}u$ are proven. Furthermore, we define a mean random dynamical system due to the uniqueness of the solution, which has a unique weak pullback mean random attractor in $L^蟻\left(惟; L^2\left(\mathbb{R}^n\right)\right)$. This result highlights the long-term dynamics of a broad class of stochastic fractional dispersion equations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.02781v1-abstract-full').style.display = 'none'; document.getElementById('2411.02781v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.22141">arXiv:2410.22141</a> <span> [<a href="https://arxiv.org/pdf/2410.22141">pdf</a>, <a href="https://arxiv.org/format/2410.22141">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Averaging principle for multiscale controlled jump diffusions and associated nonlocal HJB equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+Q">Qi Zhang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yanjie Zhang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Ao Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.22141v2-abstract-short" style="display: inline;"> This paper studies the averaging principle of a class of two-time scale stochastic control systems with $伪$-stable noise. The associated singular perturbations problem for nonlocal Hamilton-Jacobi-Bellman (HJB) equations is also considered. We construct the effective stochastic control problem and the associated effective HJB equation for the original multiscale stochastic control problem by avera… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22141v2-abstract-full').style.display = 'inline'; document.getElementById('2410.22141v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.22141v2-abstract-full" style="display: none;"> This paper studies the averaging principle of a class of two-time scale stochastic control systems with $伪$-stable noise. The associated singular perturbations problem for nonlocal Hamilton-Jacobi-Bellman (HJB) equations is also considered. We construct the effective stochastic control problem and the associated effective HJB equation for the original multiscale stochastic control problem by averaging over the ergodic measure of the fast component. We prove the convergence of the value function using two methods. With the probabilistic method, we show that the value function of the original multiscale stochastic control system converges to the value function of the effective stochastic control system by proving the weak averaging principle of the controlled jump diffusions. Using the PDE method, we study the value function as the viscosity solution of the associated nonlocal HJB equation with singular perturbations. We then prove the convergence using the perturbed test function method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22141v2-abstract-full').style.display = 'none'; document.getElementById('2410.22141v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.14046">arXiv:2410.14046</a> <span> [<a href="https://arxiv.org/pdf/2410.14046">pdf</a>, <a href="https://arxiv.org/format/2410.14046">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Tensor Decomposition with Unaligned Observations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Tang%2C+R">Runshi Tang</a>, <a href="/search/math?searchtype=author&query=Kolda%2C+T">Tamara Kolda</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.14046v1-abstract-short" style="display: inline;"> This paper presents a canonical polyadic (CP) tensor decomposition that addresses unaligned observations. The mode with unaligned observations is represented using functions in a reproducing kernel Hilbert space (RKHS). We introduce a versatile loss function that effectively accounts for various types of data, including binary, integer-valued, and positive-valued types. Additionally, we propose an… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.14046v1-abstract-full').style.display = 'inline'; document.getElementById('2410.14046v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.14046v1-abstract-full" style="display: none;"> This paper presents a canonical polyadic (CP) tensor decomposition that addresses unaligned observations. The mode with unaligned observations is represented using functions in a reproducing kernel Hilbert space (RKHS). We introduce a versatile loss function that effectively accounts for various types of data, including binary, integer-valued, and positive-valued types. Additionally, we propose an optimization algorithm for computing tensor decompositions with unaligned observations, along with a stochastic gradient method to enhance computational efficiency. A sketching algorithm is also introduced to further improve efficiency when using the $\ell_2$ loss function. To demonstrate the efficacy of our methods, we provide illustrative examples using both synthetic data and an early childhood human microbiome dataset. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.14046v1-abstract-full').style.display = 'none'; document.getElementById('2410.14046v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.06381">arXiv:2410.06381</a> <span> [<a href="https://arxiv.org/pdf/2410.06381">pdf</a>, <a href="https://arxiv.org/format/2410.06381">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Statistical Inference for Low-Rank Tensors: Heteroskedasticity, Subgaussianity, and Applications </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Agterberg%2C+J">Joshua Agterberg</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Anru Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.06381v1-abstract-short" style="display: inline;"> In this paper, we consider inference and uncertainty quantification for low Tucker rank tensors with additive noise in the high-dimensional regime. Focusing on the output of the higher-order orthogonal iteration (HOOI) algorithm, a commonly used algorithm for tensor singular value decomposition, we establish non-asymptotic distributional theory and study how to construct confidence regions and int… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06381v1-abstract-full').style.display = 'inline'; document.getElementById('2410.06381v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.06381v1-abstract-full" style="display: none;"> In this paper, we consider inference and uncertainty quantification for low Tucker rank tensors with additive noise in the high-dimensional regime. Focusing on the output of the higher-order orthogonal iteration (HOOI) algorithm, a commonly used algorithm for tensor singular value decomposition, we establish non-asymptotic distributional theory and study how to construct confidence regions and intervals for both the estimated singular vectors and the tensor entries in the presence of heteroskedastic subgaussian noise, which are further shown to be optimal for homoskedastic Gaussian noise. Furthermore, as a byproduct of our theoretical results, we establish the entrywise convergence of HOOI when initialized via diagonal deletion. To further illustrate the utility of our theoretical results, we then consider several concrete statistical inference tasks. First, in the tensor mixed-membership blockmodel, we consider a two-sample test for equality of membership profiles, and we propose a test statistic with consistency under local alternatives that exhibits a power improvement relative to the corresponding matrix test considered in several previous works. Next, we consider simultaneous inference for small collections of entries of the tensor, and we obtain consistent confidence regions. Finally, focusing on the particular case of testing whether entries of the tensor are equal, we propose a consistent test statistic that shows how index overlap results in different asymptotic standard deviations. All of our proposed procedures are fully data-driven, adaptive to noise distribution and signal strength, and do not rely on sample-splitting, and our main results highlight the effect of higher-order structures on estimation relative to the matrix setting. Our theoretical results are demonstrated through numerical simulations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06381v1-abstract-full').style.display = 'none'; document.getElementById('2410.06381v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.03619">arXiv:2410.03619</a> <span> [<a href="https://arxiv.org/pdf/2410.03619">pdf</a>, <a href="https://arxiv.org/format/2410.03619">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> </div> </div> <p class="title is-5 mathjax"> Functional Singular Value Decomposition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Tan%2C+J">Jianbin Tan</a>, <a href="/search/math?searchtype=author&query=Shi%2C+P">Pixu Shi</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.03619v4-abstract-short" style="display: inline;"> Heterogeneous functional data commonly arise in time series and longitudinal studies. To uncover the statistical structures of such data, we propose Functional Singular Value Decomposition (FSVD), a unified framework encompassing various tasks for the analysis of functional data with potential heterogeneity. We establish the mathematical foundation of FSVD by proving its existence and providing it… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.03619v4-abstract-full').style.display = 'inline'; document.getElementById('2410.03619v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.03619v4-abstract-full" style="display: none;"> Heterogeneous functional data commonly arise in time series and longitudinal studies. To uncover the statistical structures of such data, we propose Functional Singular Value Decomposition (FSVD), a unified framework encompassing various tasks for the analysis of functional data with potential heterogeneity. We establish the mathematical foundation of FSVD by proving its existence and providing its fundamental properties. We then develop an implementation approach for noisy and irregularly observed functional data based on a novel alternating minimization scheme and provide theoretical guarantees for its convergence and estimation accuracy. The FSVD framework also introduces the concepts of intrinsic basis functions and intrinsic basis vectors, representing two fundamental structural aspects of random functions. These concepts enable FSVD to provide new and improved solutions to tasks including functional principal component analysis, factor models, functional clustering, functional linear regression, and functional completion, while effectively handling heterogeneity and irregular temporal sampling. Through extensive simulations, we demonstrate that FSVD-based methods consistently outperform existing methods across these tasks. To showcase the value of FSVD in real-world datasets, we apply it to extract temporal patterns from a COVID-19 case count dataset and perform data completion on an electronic health record dataset. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.03619v4-abstract-full').style.display = 'none'; document.getElementById('2410.03619v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.17849">arXiv:2409.17849</a> <span> [<a href="https://arxiv.org/pdf/2409.17849">pdf</a>, <a href="https://arxiv.org/ps/2409.17849">ps</a>, <a href="https://arxiv.org/format/2409.17849">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Modular representation of Reductive Lie algebras and related combinatorial category </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A">An Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.17849v1-abstract-short" style="display: inline;"> We introduce and study a ``combinatorial" category related to the representations of reduced enveloping algebras of reductive Lie algebras in ``standard Levi form". It is compatible with the so-called AJS category in \cite{AJS94}, where AJS category is an important role in studying the Lusztig's conjecture on characters of irreducible modules of algebraic group. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.17849v1-abstract-full" style="display: none;"> We introduce and study a ``combinatorial" category related to the representations of reduced enveloping algebras of reductive Lie algebras in ``standard Levi form". It is compatible with the so-called AJS category in \cite{AJS94}, where AJS category is an important role in studying the Lusztig's conjecture on characters of irreducible modules of algebraic group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.17849v1-abstract-full').style.display = 'none'; document.getElementById('2409.17849v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages, comments are welcome!</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.04031">arXiv:2409.04031</a> <span> [<a href="https://arxiv.org/pdf/2409.04031">pdf</a>, <a href="https://arxiv.org/ps/2409.04031">ps</a>, <a href="https://arxiv.org/format/2409.04031">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Rate of convergence of the Kac particle system for the Boltzmann equation with hard potentials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+C">Chenguang Liu</a>, <a href="/search/math?searchtype=author&query=Xu%2C+L">Liping Xu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">An Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.04031v1-abstract-short" style="display: inline;"> In this paper, we prove that the Kac stochastic particle system converges to the weak solution of the spatially homogeneous Boltzmann equation for hard potentials and hard spheres. We give, under the initial data with finite exponential moment assumption, an explicit rate of propagation of chaos in squared Wasserstein distance with quadratic cost by using a double coupling technique. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.04031v1-abstract-full" style="display: none;"> In this paper, we prove that the Kac stochastic particle system converges to the weak solution of the spatially homogeneous Boltzmann equation for hard potentials and hard spheres. We give, under the initial data with finite exponential moment assumption, an explicit rate of propagation of chaos in squared Wasserstein distance with quadratic cost by using a double coupling technique. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04031v1-abstract-full').style.display = 'none'; document.getElementById('2409.04031v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.05677">arXiv:2408.05677</a> <span> [<a href="https://arxiv.org/pdf/2408.05677">pdf</a>, <a href="https://arxiv.org/format/2408.05677">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Tensor Decomposition Meets RKHS: Efficient Algorithms for Smooth and Misaligned Data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Larsen%2C+B+W">Brett W. Larsen</a>, <a href="/search/math?searchtype=author&query=Kolda%2C+T+G">Tamara G. Kolda</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a>, <a href="/search/math?searchtype=author&query=Williams%2C+A+H">Alex H. Williams</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.05677v1-abstract-short" style="display: inline;"> The canonical polyadic (CP) tensor decomposition decomposes a multidimensional data array into a sum of outer products of finite-dimensional vectors. Instead, we can replace some or all of the vectors with continuous functions (infinite-dimensional vectors) from a reproducing kernel Hilbert space (RKHS). We refer to tensors with some infinite-dimensional modes as quasitensors, and the approach of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05677v1-abstract-full').style.display = 'inline'; document.getElementById('2408.05677v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.05677v1-abstract-full" style="display: none;"> The canonical polyadic (CP) tensor decomposition decomposes a multidimensional data array into a sum of outer products of finite-dimensional vectors. Instead, we can replace some or all of the vectors with continuous functions (infinite-dimensional vectors) from a reproducing kernel Hilbert space (RKHS). We refer to tensors with some infinite-dimensional modes as quasitensors, and the approach of decomposing a tensor with some continuous RKHS modes is referred to as CP-HiFi (hybrid infinite and finite dimensional) tensor decomposition. An advantage of CP-HiFi is that it can enforce smoothness in the infinite dimensional modes. Further, CP-HiFi does not require the observed data to lie on a regular and finite rectangular grid and naturally incorporates misaligned data. We detail the methodology and illustrate it on a synthetic example. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05677v1-abstract-full').style.display = 'none'; document.getElementById('2408.05677v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.19741">arXiv:2407.19741</a> <span> [<a href="https://arxiv.org/pdf/2407.19741">pdf</a>, <a href="https://arxiv.org/ps/2407.19741">ps</a>, <a href="https://arxiv.org/format/2407.19741">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Scaling limits for supercritical nearly unstable Hawkes processes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+C">Chenguang Liu</a>, <a href="/search/math?searchtype=author&query=Xu%2C+L">Liping Xu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">An Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.19741v2-abstract-short" style="display: inline;"> In this paper, we investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has $L^1$ norm strictly greater than one and close to one as time goes to infinity. We find that,the scaling size determines the scaling behavior of the processes like in \cite{MR3313750}. Specifically,after suitable rescaling, the limit of the sequence of Hawkes processes is determini… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.19741v2-abstract-full').style.display = 'inline'; document.getElementById('2407.19741v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.19741v2-abstract-full" style="display: none;"> In this paper, we investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has $L^1$ norm strictly greater than one and close to one as time goes to infinity. We find that,the scaling size determines the scaling behavior of the processes like in \cite{MR3313750}. Specifically,after suitable rescaling, the limit of the sequence of Hawkes processes is deterministic.And also with another appropriate rescaling, the sequence converges in law to an integrated Cox Ingersoll Ross like process. This theoretical result may apply to model the recent COVID19 in epidemiology and in social network. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.19741v2-abstract-full').style.display = 'none'; document.getElementById('2407.19741v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.09149">arXiv:2404.09149</a> <span> [<a href="https://arxiv.org/pdf/2404.09149">pdf</a>, <a href="https://arxiv.org/format/2404.09149">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Systems and Control">eess.SY</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Neural and Evolutionary Computing">cs.NE</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Heuristic Solution to Joint Deployment and Beamforming Design for STAR-RIS Aided Networks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yan%2C+B">Bai Yan</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+Q">Qi Zhao</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jin Zhang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J+A">J. Andrew Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.09149v1-abstract-short" style="display: inline;"> This paper tackles the deployment challenges of Simultaneous Transmitting and Reflecting Reconfigurable Intelligent Surface (STAR-RIS) in communication systems. Unlike existing works that use fixed deployment setups or solely optimize the location, this paper emphasizes the joint optimization of the location and orientation of STAR-RIS. This enables searching across all user grouping possibilities… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.09149v1-abstract-full').style.display = 'inline'; document.getElementById('2404.09149v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.09149v1-abstract-full" style="display: none;"> This paper tackles the deployment challenges of Simultaneous Transmitting and Reflecting Reconfigurable Intelligent Surface (STAR-RIS) in communication systems. Unlike existing works that use fixed deployment setups or solely optimize the location, this paper emphasizes the joint optimization of the location and orientation of STAR-RIS. This enables searching across all user grouping possibilities and fully boosting the system's performance. We consider a sum rate maximization problem with joint optimization and hybrid beamforming design. An offline heuristic solution is proposed for the problem, developed based on differential evolution and semi-definite programming methods. In particular, a point-point representation is proposed for characterizing and exploiting the user-grouping. A balanced grouping method is designed to achieve a desired user grouping with low complexity. Numerical results demonstrate the substantial performance gains achievable through optimal deployment design. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.09149v1-abstract-full').style.display = 'none'; document.getElementById('2404.09149v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.16168">arXiv:2403.16168</a> <span> [<a href="https://arxiv.org/pdf/2403.16168">pdf</a>, <a href="https://arxiv.org/ps/2403.16168">ps</a>, <a href="https://arxiv.org/format/2403.16168">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1017/fms.2024.112">10.1017/fms.2024.112 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantum bumpless pipe dreams </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Le%2C+T">Tuong Le</a>, <a href="/search/math?searchtype=author&query=Ouyang%2C+S">Shuge Ouyang</a>, <a href="/search/math?searchtype=author&query=Tao%2C+L">Leo Tao</a>, <a href="/search/math?searchtype=author&query=Restivo%2C+J">Joseph Restivo</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Angelina Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.16168v2-abstract-short" style="display: inline;"> Schubert polynomials are polynomial representatives of Schubert classes in the cohomology of the complete flag variety and have a combinatorial formulation in terms of bumpless pipe dreams. Quantum double Schubert polynomials are polynomial representatives of Schubert classes in the torus-equivariant quantum cohomology of the complete flag variety, but no analogous combinatorial formulation had be… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.16168v2-abstract-full').style.display = 'inline'; document.getElementById('2403.16168v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.16168v2-abstract-full" style="display: none;"> Schubert polynomials are polynomial representatives of Schubert classes in the cohomology of the complete flag variety and have a combinatorial formulation in terms of bumpless pipe dreams. Quantum double Schubert polynomials are polynomial representatives of Schubert classes in the torus-equivariant quantum cohomology of the complete flag variety, but no analogous combinatorial formulation had been discovered. We introduce a generalization of the bumpless pipe dreams called quantum bumpless pipe dreams, giving a novel combinatorial formula for quantum double Schubert polynomials as a sum of binomial weights of quantum bumpless pipe dreams. We give a bijective proof for this formula by showing that the sum of binomial weights satisfies a defining transition equation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.16168v2-abstract-full').style.display = 'none'; document.getElementById('2403.16168v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages, 20 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Forum of Mathematics, Sigma 13 (2025) e28 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.15572">arXiv:2403.15572</a> <span> [<a href="https://arxiv.org/pdf/2403.15572">pdf</a>, <a href="https://arxiv.org/ps/2403.15572">ps</a>, <a href="https://arxiv.org/format/2403.15572">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> </div> <p class="title is-5 mathjax"> Bounding the $K(p-1)$-local exotic Picard group at $p>3$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bobkova%2C+I">Irina Bobkova</a>, <a href="/search/math?searchtype=author&query=Lachmann%2C+A">Andrea Lachmann</a>, <a href="/search/math?searchtype=author&query=Li%2C+A">Ang Li</a>, <a href="/search/math?searchtype=author&query=Lima%2C+A">Alicia Lima</a>, <a href="/search/math?searchtype=author&query=Stojanoska%2C+V">Vesna Stojanoska</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+Y">Adela YiYu Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.15572v2-abstract-short" style="display: inline;"> In this paper, we bound the descent filtration of the exotic Picard group $魏_n$, for a prime number p>3 and n=p-1. Our method involves a detailed comparison of the Picard spectral sequence, the homotopy fixed point spectral sequence, and an auxiliary $尾$-inverted homotopy fixed point spectral sequence whose input is the Farrell-Tate cohomology of the Morava stabilizer group. Along the way, we dedu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.15572v2-abstract-full').style.display = 'inline'; document.getElementById('2403.15572v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.15572v2-abstract-full" style="display: none;"> In this paper, we bound the descent filtration of the exotic Picard group $魏_n$, for a prime number p>3 and n=p-1. Our method involves a detailed comparison of the Picard spectral sequence, the homotopy fixed point spectral sequence, and an auxiliary $尾$-inverted homotopy fixed point spectral sequence whose input is the Farrell-Tate cohomology of the Morava stabilizer group. Along the way, we deduce that the K(n)-local Adams-Novikov spectral sequence for the sphere has a horizontal vanishing line at $3n^2+1$ on the $E_{2n^2+2}$-page. The same analysis also allows us to express the exotic Picard group of $K(n)$-local modules over the homotopy fixed points spectrum $\mathrm{E}_n^{hN}$, where N is the normalizer in $\mathbb{G}_n$ of a finite cyclic subgroup of order p, as a subquotient of a single continuous cohomology group $H^{2n+1}(N,蟺_{2n}\mathrm{E}_n)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.15572v2-abstract-full').style.display = 'none'; document.getElementById('2403.15572v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> CPH-GEOTOP-DNRF151 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.15608">arXiv:2401.15608</a> <span> [<a href="https://arxiv.org/pdf/2401.15608">pdf</a>, <a href="https://arxiv.org/ps/2401.15608">ps</a>, <a href="https://arxiv.org/format/2401.15608">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> The stochastic fractional nonlinear Schr枚dinger equations in $H^伪$ and structure-preserving algorithm </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A">Ao Zhang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yanjie Zhang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+P">Pengde Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+X">Xiao Wang</a>, <a href="/search/math?searchtype=author&query=Duan%2C+J">Jinqiao Duan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.15608v2-abstract-short" style="display: inline;"> In this paper, we first investigate the global existence of a solution for the stochastic fractional nonlinear Schr枚dinger equation with radially symmetric initial data in a suitable energy space $H^伪$. We then show that the stochastic fractional nonlinear Schr枚dinger equation in the Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system, with its phase flow preserving symp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.15608v2-abstract-full').style.display = 'inline'; document.getElementById('2401.15608v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.15608v2-abstract-full" style="display: none;"> In this paper, we first investigate the global existence of a solution for the stochastic fractional nonlinear Schr枚dinger equation with radially symmetric initial data in a suitable energy space $H^伪$. We then show that the stochastic fractional nonlinear Schr枚dinger equation in the Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system, with its phase flow preserving symplecticity. Finally, we develop a stochastic midpoint scheme for the stochastic fractional nonlinear Schr枚dinger equation from the perspective of symplectic geometry. It is proved that the stochastic midpoint scheme satisfies the corresponding symplectic law in the discrete sense. A numerical example is conducted to validate the efficiency of the theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.15608v2-abstract-full').style.display = 'none'; document.getElementById('2401.15608v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.06214">arXiv:2312.06214</a> <span> [<a href="https://arxiv.org/pdf/2312.06214">pdf</a>, <a href="https://arxiv.org/ps/2312.06214">ps</a>, <a href="https://arxiv.org/format/2312.06214">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Duplex Hecke Algebras of type B </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+Y">Yu Xie</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">An Zhang</a>, <a href="/search/math?searchtype=author&query=Shu%2C+B">Bin Shu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.06214v2-abstract-short" style="display: inline;"> As a sequel to [14], in this article we first introduce a so-called duplex Hecke algebras of type B which is a Q(q)-algebra associated with the Weyl group W (B) of type B, and symmetric groups S_l for l = 0, 1, . . . ,m, satisfying some Hecke relations. This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur-Weyl duality of Levi-type, ext… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.06214v2-abstract-full').style.display = 'inline'; document.getElementById('2312.06214v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.06214v2-abstract-full" style="display: none;"> As a sequel to [14], in this article we first introduce a so-called duplex Hecke algebras of type B which is a Q(q)-algebra associated with the Weyl group W (B) of type B, and symmetric groups S_l for l = 0, 1, . . . ,m, satisfying some Hecke relations. This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur-Weyl duality of Levi-type, extending the duplex Hecke algebra of type A arising from the related q-Schur-Weyl duality of Levi-type. A duplex Hecke algebra of type B admits natural representations on certain tensor spaces. We then establish a Levi-type q-Schur-Weyl duality of type B, which reveals the double centralizer property between such duplex Hecke algebras and 谋quantum groups studied by Bao-Wang in [1]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.06214v2-abstract-full').style.display = 'none'; document.getElementById('2312.06214v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages. To appear in Journal of Algebra and its Applications</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G05; 17B20; 17B45; 17B50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.11834">arXiv:2311.11834</a> <span> [<a href="https://arxiv.org/pdf/2311.11834">pdf</a>, <a href="https://arxiv.org/format/2311.11834">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Congressional Districting: "Rocks-Pebbles-Sand" </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Risk%2C+J">Jimmy Risk</a>, <a href="/search/math?searchtype=author&query=Switkes%2C+J">Jennifer Switkes</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Ann Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.11834v1-abstract-short" style="display: inline;"> As a case study into an algorithmic approach to congressional districting, North Carolina provides a lot to explore. Statistical modeling has called into question whether recent North Carolina district plans are unbiased. In particular, the literature suggests that the district plan used in the 2016 U.S. House of Representatives election yields outlier results that are statistically unlikely to be… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.11834v1-abstract-full').style.display = 'inline'; document.getElementById('2311.11834v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.11834v1-abstract-full" style="display: none;"> As a case study into an algorithmic approach to congressional districting, North Carolina provides a lot to explore. Statistical modeling has called into question whether recent North Carolina district plans are unbiased. In particular, the literature suggests that the district plan used in the 2016 U.S. House of Representatives election yields outlier results that are statistically unlikely to be obtained without the application of bias. Therefore, methods for creating strong and fair district plans are needed. Informed by previous districting models and algorithms, we build a model and algorithm to produce an ensemble of viable Congressional district plans. Our work contributes a ``Rocks-Pebbles-Sand'' concept and procedure facilitating reasonable population equity with a small overall number of county splits among districts. Additionally, our methodology minimizes the initial need for granular, precinct-level data, thereby reducing the risk of covert gerrymandering. This case study indicates plausibility of an approach built upon an easy-to-understand intuition. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.11834v1-abstract-full').style.display = 'none'; document.getElementById('2311.11834v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages, 14 figures, 10 tables</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.07773">arXiv:2311.07773</a> <span> [<a href="https://arxiv.org/pdf/2311.07773">pdf</a>, <a href="https://arxiv.org/ps/2311.07773">ps</a>, <a href="https://arxiv.org/format/2311.07773">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Computational and Statistical Thresholds in Multi-layer Stochastic Block Models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lei%2C+J">Jing Lei</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+Z">Zihan Zhu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.07773v1-abstract-short" style="display: inline;"> We study the problem of community recovery and detection in multi-layer stochastic block models, focusing on the critical network density threshold for consistent community structure inference. Using a prototypical two-block model, we reveal a computational barrier for such multi-layer stochastic block models that does not exist for its single-layer counterpart: When there are no computational con… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.07773v1-abstract-full').style.display = 'inline'; document.getElementById('2311.07773v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.07773v1-abstract-full" style="display: none;"> We study the problem of community recovery and detection in multi-layer stochastic block models, focusing on the critical network density threshold for consistent community structure inference. Using a prototypical two-block model, we reveal a computational barrier for such multi-layer stochastic block models that does not exist for its single-layer counterpart: When there are no computational constraints, the density threshold depends linearly on the number of layers. However, when restricted to polynomial-time algorithms, the density threshold scales with the square root of the number of layers, assuming correctness of a low-degree polynomial hardness conjecture. Our results provide a nearly complete picture of the optimal inference in multiple-layer stochastic block models and partially settle the open question in Lei and Lin (2022) regarding the optimality of the bias-adjusted spectral method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.07773v1-abstract-full').style.display = 'none'; document.getElementById('2311.07773v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 62C20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.03511">arXiv:2311.03511</a> <span> [<a href="https://arxiv.org/pdf/2311.03511">pdf</a>, <a href="https://arxiv.org/format/2311.03511">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Convergence from the discrete to the continuous non-linear Fourier transform </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Ashley R. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.03511v2-abstract-short" style="display: inline;"> In this note, we study the convergence from the discrete to the continuous non-linear Fourier transform. Relations between spectral problems and questions in complex function theory provide a new approach to the study of scattering problems and the non-linear Fourier transform \cite{Scatter}. In particular, the non-linear Fourier transform can be viewed from the perspective of spectral problems fo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.03511v2-abstract-full').style.display = 'inline'; document.getElementById('2311.03511v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.03511v2-abstract-full" style="display: none;"> In this note, we study the convergence from the discrete to the continuous non-linear Fourier transform. Relations between spectral problems and questions in complex function theory provide a new approach to the study of scattering problems and the non-linear Fourier transform \cite{Scatter}. In particular, the non-linear Fourier transform can be viewed from the perspective of spectral problems for differential operators. Results in \cite{MP, PZ} can be seen as results for the non-linear Fourier transform. These results are similar to some convergence problems for the discrete non-linear Fourier transform considered in \cite{T} and \cite{TT}. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.03511v2-abstract-full').style.display = 'none'; document.getElementById('2311.03511v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.02470">arXiv:2311.02470</a> <span> [<a href="https://arxiv.org/pdf/2311.02470">pdf</a>, <a href="https://arxiv.org/ps/2311.02470">ps</a>, <a href="https://arxiv.org/format/2311.02470">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Gradient estimates and Liouville theorems for Lichnerowicz-type equation on Riemannian manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Y">Youde Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Aiqi Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.02470v2-abstract-short" style="display: inline;"> In this paper we consider the gradient estimates on positive solutions to the following elliptic (Lichnerowicz) equation defined on a complete Riemannian manifold $(M,\,g)$: $$螖v + 渭v + a v^{p+1} +b v^{-q+1} =0,$$ where $p\geq-1$, $q\geq1$, $渭$, $a$ and $b$ are real constants. In the case $渭\geq0$ and $b\geq0$ or $渭<0$ , $a>0$ and $b>0$ ($渭$ has a lower bound), we employ the Nash-Moser iteration… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.02470v2-abstract-full').style.display = 'inline'; document.getElementById('2311.02470v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.02470v2-abstract-full" style="display: none;"> In this paper we consider the gradient estimates on positive solutions to the following elliptic (Lichnerowicz) equation defined on a complete Riemannian manifold $(M,\,g)$: $$螖v + 渭v + a v^{p+1} +b v^{-q+1} =0,$$ where $p\geq-1$, $q\geq1$, $渭$, $a$ and $b$ are real constants. In the case $渭\geq0$ and $b\geq0$ or $渭<0$ , $a>0$ and $b>0$ ($渭$ has a lower bound), we employ the Nash-Moser iteration technique to obtain some refined gradient estimates of the solutions to the above equation, if $(M,\,g)$ satisfies $Ric \geq -(n-1)魏$ , where $n\geq3$ is the dimension of $M$ and $魏$ is a nonnegative constant, and $渭$ , $a$ , $b$ , $p$ and $q$ satisfy some technique conditions. By the obtained gradient estimates we also derive some Liouville type theorems for the above equation under some suitable geometric and analysis conditions. As applications, we can derive some Cheng-Yau's type gradient estimates for solutions to the $n$-dimensional Einstein-scalar field Lichnerowicz equation where $n\geq3$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.02470v2-abstract-full').style.display = 'none'; document.getElementById('2311.02470v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: text overlap with arXiv:2309.05367</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.09103">arXiv:2309.09103</a> <span> [<a href="https://arxiv.org/pdf/2309.09103">pdf</a>, <a href="https://arxiv.org/format/2309.09103">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Econometrics">econ.EM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Optimal Estimation under a Semiparametric Density Ratio Model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A+G">Archer Gong Zhang</a>, <a href="/search/math?searchtype=author&query=Chen%2C+J">Jiahua Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.09103v1-abstract-short" style="display: inline;"> In many statistical and econometric applications, we gather individual samples from various interconnected populations that undeniably exhibit common latent structures. Utilizing a model that incorporates these latent structures for such data enhances the efficiency of inferences. Recently, many researchers have been adopting the semiparametric density ratio model (DRM) to address the presence of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.09103v1-abstract-full').style.display = 'inline'; document.getElementById('2309.09103v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.09103v1-abstract-full" style="display: none;"> In many statistical and econometric applications, we gather individual samples from various interconnected populations that undeniably exhibit common latent structures. Utilizing a model that incorporates these latent structures for such data enhances the efficiency of inferences. Recently, many researchers have been adopting the semiparametric density ratio model (DRM) to address the presence of latent structures. The DRM enables estimation of each population distribution using pooled data, resulting in statistically more efficient estimations in contrast to nonparametric methods that analyze each sample in isolation. In this article, we investigate the limit of the efficiency improvement attainable through the DRM. We focus on situations where one population's sample size significantly exceeds those of the other populations. In such scenarios, we demonstrate that the DRM-based inferences for populations with smaller sample sizes achieve the highest attainable asymptotic efficiency as if a parametric model is assumed. The estimands we consider include the model parameters, distribution functions, and quantiles. We use simulation experiments to support the theoretical findings with a specific focus on quantile estimation. Additionally, we provide an analysis of real revenue data from U.S. collegiate sports to illustrate the efficacy of our contribution. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.09103v1-abstract-full').style.display = 'none'; document.getElementById('2309.09103v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.05367">arXiv:2309.05367</a> <span> [<a href="https://arxiv.org/pdf/2309.05367">pdf</a>, <a href="https://arxiv.org/ps/2309.05367">ps</a>, <a href="https://arxiv.org/format/2309.05367">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Gradient Estimate for Solutions of $螖v+v^r-v^s= 0$ on A Complete Riemannian Manifold </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Y">Youde Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Aiqi Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.05367v3-abstract-short" style="display: inline;"> In this paper we consider the gradient estimates on positive solutions to the following elliptic equation defined on a complete Riemannian manifold $(M,\,g)$: $$螖v+v^r-v^s= 0,$$ where $r$ and $s$ are two real constants. When$(M,\,g)$ satisfies $Ric \geq -(n-1)魏$ (where $n\geq2$ is the dimension of $M$ and $魏$ is a nonnegative constant), we employ the Nash-Moser iteration technique to derive a Ch… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.05367v3-abstract-full').style.display = 'inline'; document.getElementById('2309.05367v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.05367v3-abstract-full" style="display: none;"> In this paper we consider the gradient estimates on positive solutions to the following elliptic equation defined on a complete Riemannian manifold $(M,\,g)$: $$螖v+v^r-v^s= 0,$$ where $r$ and $s$ are two real constants. When$(M,\,g)$ satisfies $Ric \geq -(n-1)魏$ (where $n\geq2$ is the dimension of $M$ and $魏$ is a nonnegative constant), we employ the Nash-Moser iteration technique to derive a Cheng-Yau's type gradient estimate for positive solution to the above equation under some suitable geometric and analysis conditions. Moreover, it is shown that when the Ricci curvature of $M$ is nonnegative, this elliptic equation does not admit any positive solution except for $u\equiv 1$ if $r<s$ and $$1<r<\frac{n+3}{n-1}\quad\quad ~~\mbox{or}~~\quad 1<s<\frac{n+3}{n-1}.$$ <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.05367v3-abstract-full').style.display = 'none'; document.getElementById('2309.05367v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.10270">arXiv:2308.10270</a> <span> [<a href="https://arxiv.org/pdf/2308.10270">pdf</a>, <a href="https://arxiv.org/ps/2308.10270">ps</a>, <a href="https://arxiv.org/format/2308.10270">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> The stochastic fractional Strichartz estimate and blow-up for Schr枚dinger equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A">Ao Zhang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yanjie Zhang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+X">Xiao Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Z">Zibo Wang</a>, <a href="/search/math?searchtype=author&query=Duan%2C+J">Jinqiao Duan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2308.10270v3-abstract-short" style="display: inline;"> We establish the stochastic Strichartz estimate for the fractional Schr枚dinger equation with multiplicative noise. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to the stochastic fractional nonlinear Schr枚dinger equation in $L^2(\mathbb{R}^n)$. In addition, we also prove a general blow up result by deriving a localized virial es… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.10270v3-abstract-full').style.display = 'inline'; document.getElementById('2308.10270v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.10270v3-abstract-full" style="display: none;"> We establish the stochastic Strichartz estimate for the fractional Schr枚dinger equation with multiplicative noise. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to the stochastic fractional nonlinear Schr枚dinger equation in $L^2(\mathbb{R}^n)$. In addition, we also prove a general blow up result by deriving a localized virial estimate and the generalized Strauss inequality with a restricted class of initial data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.10270v3-abstract-full').style.display = 'none'; document.getElementById('2308.10270v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.11597">arXiv:2307.11597</a> <span> [<a href="https://arxiv.org/pdf/2307.11597">pdf</a>, <a href="https://arxiv.org/ps/2307.11597">ps</a>, <a href="https://arxiv.org/format/2307.11597">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Improved Spectral Cluster Bounds for Orthonormal Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ren%2C+T">Tianyi Ren</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">An Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.11597v1-abstract-short" style="display: inline;"> We improve Frank-Sabin's work concerning the spectral cluster bounds for orthonormal systems at $p=\infty$, on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from $[位^{2}, (位+1)^{2})$ to $[位^{2}, (位+蔚(位))^{2})$, where $蔚(位)$ is a function of $位$ that goes to $0$ as $位$ goes to $\infty$. In achieving this, we invoke the method developed by Bourgain-Shao… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.11597v1-abstract-full').style.display = 'inline'; document.getElementById('2307.11597v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.11597v1-abstract-full" style="display: none;"> We improve Frank-Sabin's work concerning the spectral cluster bounds for orthonormal systems at $p=\infty$, on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from $[位^{2}, (位+1)^{2})$ to $[位^{2}, (位+蔚(位))^{2})$, where $蔚(位)$ is a function of $位$ that goes to $0$ as $位$ goes to $\infty$. In achieving this, we invoke the method developed by Bourgain-Shao-Sogge-Yao. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.11597v1-abstract-full').style.display = 'none'; document.getElementById('2307.11597v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">12 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58J50; 35P15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.00575">arXiv:2307.00575</a> <span> [<a href="https://arxiv.org/pdf/2307.00575">pdf</a>, <a href="https://arxiv.org/format/2307.00575">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Mode-wise Principal Subspace Pursuit and Matrix Spiked Covariance Model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Tang%2C+R">Runshi Tang</a>, <a href="/search/math?searchtype=author&query=Yuan%2C+M">Ming Yuan</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.00575v2-abstract-short" style="display: inline;"> This paper introduces a novel framework called Mode-wise Principal Subspace Pursuit (MOP-UP) to extract hidden variations in both the row and column dimensions for matrix data. To enhance the understanding of the framework, we introduce a class of matrix-variate spiked covariance models that serve as inspiration for the development of the MOP-UP algorithm. The MOP-UP algorithm consists of two step… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.00575v2-abstract-full').style.display = 'inline'; document.getElementById('2307.00575v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.00575v2-abstract-full" style="display: none;"> This paper introduces a novel framework called Mode-wise Principal Subspace Pursuit (MOP-UP) to extract hidden variations in both the row and column dimensions for matrix data. To enhance the understanding of the framework, we introduce a class of matrix-variate spiked covariance models that serve as inspiration for the development of the MOP-UP algorithm. The MOP-UP algorithm consists of two steps: Average Subspace Capture (ASC) and Alternating Projection (AP). These steps are specifically designed to capture the row-wise and column-wise dimension-reduced subspaces which contain the most informative features of the data. ASC utilizes a novel average projection operator as initialization and achieves exact recovery in the noiseless setting. We analyze the convergence and non-asymptotic error bounds of MOP-UP, introducing a blockwise matrix eigenvalue perturbation bound that proves the desired bound, where classic perturbation bounds fail. The effectiveness and practical merits of the proposed framework are demonstrated through experiments on both simulated and real datasets. Lastly, we discuss generalizations of our approach to higher-order data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.00575v2-abstract-full').style.display = 'none'; document.getElementById('2307.00575v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Journal of the Royal Statistical Society, Series B, to appear</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.00483">arXiv:2307.00483</a> <span> [<a href="https://arxiv.org/pdf/2307.00483">pdf</a>, <a href="https://arxiv.org/ps/2307.00483">ps</a>, <a href="https://arxiv.org/format/2307.00483">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Modular representations of strange classical Lie superalgebras and the first super Kac-Weisfeiler conjecture </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ren%2C+Y">Ye Ren</a>, <a href="/search/math?searchtype=author&query=Shu%2C+B">Bin Shu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+F">Fanlei Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">An Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.00483v1-abstract-short" style="display: inline;"> Suppose $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1} is a Lie superalgebra of queer type or periplectic type over an algebraically closed field $\textbf{k}$ of characteristic $p>2… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.00483v1-abstract-full').style.display = 'inline'; document.getElementById('2307.00483v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.00483v1-abstract-full" style="display: none;"> Suppose $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1} is a Lie superalgebra of queer type or periplectic type over an algebraically closed field $\textbf{k}$ of characteristic $p>2$. In this article, we initiate preliminarily to investigate modular representations of periplectic Lie superalgebras and then verify the first super Kac-Weisfeiler conjecture on the maximal dimensions of irreducible modules for $\mathfrak{g}$ proposed by the second-named author in [Shu] where the conjecture is targeted at all finite-dimensional restricted Lie superalgebras over $\bk$, and already proved to be true for basic classical Lie superalgebras and completely solvable restricted Lie superalgebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.00483v1-abstract-full').style.display = 'none'; document.getElementById('2307.00483v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages. Comments are welcome</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.12051">arXiv:2303.12051</a> <span> [<a href="https://arxiv.org/pdf/2303.12051">pdf</a>, <a href="https://arxiv.org/format/2303.12051">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computer Vision and Pattern Recognition">cs.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> A Novel and Optimal Spectral Method for Permutation Synchronization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Nguyen%2C+D">Duc Nguyen</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+Y">Anderson Ye Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.12051v2-abstract-short" style="display: inline;"> Permutation synchronization is an important problem in computer science that constitutes the key step of many computer vision tasks. The goal is to recover $n$ latent permutations from their noisy and incomplete pairwise measurements. In recent years, spectral methods have gained increasing popularity thanks to their simplicity and computational efficiency. Spectral methods utilize the leading eig… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.12051v2-abstract-full').style.display = 'inline'; document.getElementById('2303.12051v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.12051v2-abstract-full" style="display: none;"> Permutation synchronization is an important problem in computer science that constitutes the key step of many computer vision tasks. The goal is to recover $n$ latent permutations from their noisy and incomplete pairwise measurements. In recent years, spectral methods have gained increasing popularity thanks to their simplicity and computational efficiency. Spectral methods utilize the leading eigenspace $U$ of the data matrix and its block submatrices $U_1,U_2,\ldots, U_n$ to recover the permutations. In this paper, we propose a novel and statistically optimal spectral algorithm. Unlike the existing methods which use $\{U_jU_1^\top\}_{j\geq 2}$, ours constructs an anchor matrix $M$ by aggregating useful information from all of the block submatrices and estimates the latent permutations through $\{U_jM^\top\}_{j\geq 1}$. This modification overcomes a crucial limitation of the existing methods caused by the repetitive use of $U_1$ and leads to an improved numerical performance. To establish the optimality of the proposed method, we carry out a fine-grained spectral analysis and obtain a sharp exponential error bound that matches the minimax rate. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.12051v2-abstract-full').style.display = 'none'; document.getElementById('2303.12051v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.05024">arXiv:2303.05024</a> <span> [<a href="https://arxiv.org/pdf/2303.05024">pdf</a>, <a href="https://arxiv.org/format/2303.05024">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Social and Information Networks">cs.SI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Phase transition for detecting a small community in a large network </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jin%2C+J">Jiashun Jin</a>, <a href="/search/math?searchtype=author&query=Ke%2C+Z+T">Zheng Tracy Ke</a>, <a href="/search/math?searchtype=author&query=Turner%2C+P">Paxton Turner</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.05024v1-abstract-short" style="display: inline;"> How to detect a small community in a large network is an interesting problem, including clique detection as a special case, where a naive degree-based $蠂^2$-test was shown to be powerful in the presence of an Erd艖s-Renyi background. Using Sinkhorn's theorem, we show that the signal captured by the $蠂^2$-test may be a modeling artifact, and it may disappear once we replace the Erd艖s-Renyi model by… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.05024v1-abstract-full').style.display = 'inline'; document.getElementById('2303.05024v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.05024v1-abstract-full" style="display: none;"> How to detect a small community in a large network is an interesting problem, including clique detection as a special case, where a naive degree-based $蠂^2$-test was shown to be powerful in the presence of an Erd艖s-Renyi background. Using Sinkhorn's theorem, we show that the signal captured by the $蠂^2$-test may be a modeling artifact, and it may disappear once we replace the Erd艖s-Renyi model by a broader network model. We show that the recent SgnQ test is more appropriate for such a setting. The test is optimal in detecting communities with sizes comparable to the whole network, but has never been studied for our setting, which is substantially different and more challenging. Using a degree-corrected block model (DCBM), we establish phase transitions of this testing problem concerning the size of the small community and the edge densities in small and large communities. When the size of the small community is larger than $\sqrt{n}$, the SgnQ test is optimal for it attains the computational lower bound (CLB), the information lower bound for methods allowing polynomial computation time. When the size of the small community is smaller than $\sqrt{n}$, we establish the parameter regime where the SgnQ test has full power and make some conjectures of the CLB. We also study the classical information lower bound (LB) and show that there is always a gap between the CLB and LB in our range of interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.05024v1-abstract-full').style.display = 'none'; document.getElementById('2303.05024v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2302.02311">arXiv:2302.02311</a> <span> [<a href="https://arxiv.org/pdf/2302.02311">pdf</a>, <a href="https://arxiv.org/ps/2302.02311">ps</a>, <a href="https://arxiv.org/format/2302.02311">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Extremal values for Steiner distances and the Steiner $k$-Wiener index </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+H">Hua Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Andrew Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2302.02311v1-abstract-short" style="display: inline;"> Various questions related to distances between vertices of simple, finite graphs are of interest to extremal graph theorists. The Steiner distance of a set of $k$ vertices is a natural generalization of the regular distance. We extend several theorems on the middle parts and extremal values of trees from their regular distance variants to their Steiner distance variants. More specifically, we show… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.02311v1-abstract-full').style.display = 'inline'; document.getElementById('2302.02311v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.02311v1-abstract-full" style="display: none;"> Various questions related to distances between vertices of simple, finite graphs are of interest to extremal graph theorists. The Steiner distance of a set of $k$ vertices is a natural generalization of the regular distance. We extend several theorems on the middle parts and extremal values of trees from their regular distance variants to their Steiner distance variants. More specifically, we show that for a tree $T$, the Steiner $k$-distance, Steiner $k$-leaf-distance, and Steiner $k$-internal-distance are all concave along a path. We also calculate distances between the Steiner $k$-median, Steiner $k$-internal-median, and Steiner $k$-leaf-median. Letting the Steiner $k$-distance of a vertex $v \in V(T)$ be $\dd_k^T(v)$, we find bounds based on the order of $T$ for the ratios $\frac{\dd^T_{k}(u)}{\dd^T_{k}(v)}$, $\frac{\dd^T_{k}(w)}{\dd^T_{k}(z)}$, and $\frac{\dd^T_{k}(u)}{\dd^T_{k}(y)}$ where $u$ and $v$ are leaves, $w$ and $z$ are internal vertices, and $y$ is a Steiner $k$-centroid. Also, denoting the Steiner $k$-Wiener index as $\mathsf{SW}_k(T)$, we find upper and lower bounds for $\frac{\mathsf{SW}_k(T)}{\dd^G_{k}(v)}$. The extremal graphs that produce these bounds are also presented. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.02311v1-abstract-full').style.display = 'none'; document.getElementById('2302.02311v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2301.09289">arXiv:2301.09289</a> <span> [<a href="https://arxiv.org/pdf/2301.09289">pdf</a>, <a href="https://arxiv.org/ps/2301.09289">ps</a>, <a href="https://arxiv.org/format/2301.09289">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Social and Information Networks">cs.SI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> Fundamental Limits of Spectral Clustering in Stochastic Block Models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A+Y">Anderson Ye Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2301.09289v3-abstract-short" style="display: inline;"> Spectral clustering has been widely used for community detection in network sciences. While its empirical successes are well-documented, a clear theoretical understanding, particularly for sparse networks where degrees are much smaller than $\log n$, remains unclear. In this paper, we address this significant gap by demonstrating that spectral clustering offers exponentially small error rates when… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.09289v3-abstract-full').style.display = 'inline'; document.getElementById('2301.09289v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2301.09289v3-abstract-full" style="display: none;"> Spectral clustering has been widely used for community detection in network sciences. While its empirical successes are well-documented, a clear theoretical understanding, particularly for sparse networks where degrees are much smaller than $\log n$, remains unclear. In this paper, we address this significant gap by demonstrating that spectral clustering offers exponentially small error rates when applied to sparse networks under Stochastic Block Models. Our analysis provides sharp characterizations of its performance, backed by matching upper and lower bounds possessing an identical exponent with the same leading constant. The key to our results is a novel truncated $\ell_2$ perturbation analysis for eigenvectors, coupled with a new analysis idea of eigenvectors truncation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.09289v3-abstract-full').style.display = 'none'; document.getElementById('2301.09289v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 January, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.08642">arXiv:2212.08642</a> <span> [<a href="https://arxiv.org/pdf/2212.08642">pdf</a>, <a href="https://arxiv.org/format/2212.08642">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Estimating Higher-Order Mixed Memberships via the $\ell_{2,\infty}$ Tensor Perturbation Bound </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Agterberg%2C+J">Joshua Agterberg</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Anru Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.08642v3-abstract-short" style="display: inline;"> Higher-order multiway data is ubiquitous in machine learning and statistics and often exhibits community-like structures, where each component (node) along each different mode has a community membership associated with it. In this paper we propose the tensor mixed-membership blockmodel, a generalization of the tensor blockmodel positing that memberships need not be discrete, but instead are convex… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.08642v3-abstract-full').style.display = 'inline'; document.getElementById('2212.08642v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.08642v3-abstract-full" style="display: none;"> Higher-order multiway data is ubiquitous in machine learning and statistics and often exhibits community-like structures, where each component (node) along each different mode has a community membership associated with it. In this paper we propose the tensor mixed-membership blockmodel, a generalization of the tensor blockmodel positing that memberships need not be discrete, but instead are convex combinations of latent communities. We establish the identifiability of our model and propose a computationally efficient estimation procedure based on the higher-order orthogonal iteration algorithm (HOOI) for tensor SVD composed with a simplex corner-finding algorithm. We then demonstrate the consistency of our estimation procedure by providing a per-node error bound, which showcases the effect of higher-order structures on estimation accuracy. To prove our consistency result, we develop the $\ell_{2,\infty}$ tensor perturbation bound for HOOI under independent, heteroskedastic, subgaussian noise that may be of independent interest. Our analysis uses a novel leave-one-out construction for the iterates, and our bounds depend only on spectral properties of the underlying low-rank tensor under nearly optimal signal-to-noise ratio conditions such that tensor SVD is computationally feasible. Finally, we apply our methodology to real and simulated data, demonstrating some effects not identifiable from the model with discrete community memberships. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.08642v3-abstract-full').style.display = 'none'; document.getElementById('2212.08642v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.04170">arXiv:2212.04170</a> <span> [<a href="https://arxiv.org/pdf/2212.04170">pdf</a>, <a href="https://arxiv.org/format/2212.04170">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> An Extension of De Giorgi Class and Applications </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+H">Hongya Gao</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Aiping Zhang</a>, <a href="/search/math?searchtype=author&query=Gao%2C+S">Siyu Gao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.04170v1-abstract-short" style="display: inline;"> We present an extension of the classical De Giorgi class, and then we show that functions in this new class are locally bounded and locally H枚lder continuous. Some applications are given. As a first application, we give a regularity result for local minimizers $u:惟\subset \mathbb R^4 \rightarrow \mathbb R^4$ of a special class of polyconvex functionals with splitting form in four dimensional Eucli… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.04170v1-abstract-full').style.display = 'inline'; document.getElementById('2212.04170v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.04170v1-abstract-full" style="display: none;"> We present an extension of the classical De Giorgi class, and then we show that functions in this new class are locally bounded and locally H枚lder continuous. Some applications are given. As a first application, we give a regularity result for local minimizers $u:惟\subset \mathbb R^4 \rightarrow \mathbb R^4$ of a special class of polyconvex functionals with splitting form in four dimensional Euclidean spaces. Under some structural conditions on the energy density, we prove that each component $u^伪$ of the local minimizer $u$ belongs to the generalized De Giorgi class, then one can derive that it is locally bounded and locally H枚lder continuous. Our result can be applied to polyconvex integrals whose prototype is $$ \int_惟\Big(\sum_{伪=1}^4 |Du^伪|^p + \sum_{尾=1}^6 |({\rm adj}_2 Du )^尾| ^q +\sum_{纬=1}^4 |({\rm adj}_3 Du )^纬| ^r +|\det Du|^s \Big ) \mathrm {d}x $$ with suitable $p,q,r,s\ge 1$. As a second application, we consider a degenerate linear elliptic equation of the form $$ -\mbox {div} (a(x)\nabla u)=-\mbox {div}F, $$ with $0<a(x) \le 尾<+\infty$. We prove, by virtue of the generalized De Giorgi class, that any weak solution is locally bounded and locally H枚lder continuous provided that $\frac 1 {a(x)}$ and $F(x)$ belong to some suitable locally integrable function spaces. As a third application, we show that our theorem can be applied in dealing with regularity issues of elliptic equations with non-standard grow conditions. As a fourth application we treat with quasilinear elliptic systems. Under suitable assumptions on the coefficients, we show that any of its weak solutions is locally bounded and locally H枚lder continuous. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.04170v1-abstract-full').style.display = 'none'; document.getElementById('2212.04170v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">41 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35J20 (Primary) 35J25; 35J47 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2211.00782">arXiv:2211.00782</a> <span> [<a href="https://arxiv.org/pdf/2211.00782">pdf</a>, <a href="https://arxiv.org/ps/2211.00782">ps</a>, <a href="https://arxiv.org/format/2211.00782">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> </div> <p class="title is-5 mathjax"> Inertia groups of $(n-1)$-connected $2n$-manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Senger%2C+A">Andrew Senger</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+Y">Adela YiYu Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2211.00782v2-abstract-short" style="display: inline;"> In this paper, we compute the inertia groups of $(n-1)$-connected, smooth, closed, oriented $2n$-manifolds where $n \geq 3$. As a consequence, we complete the diffeomorphism classification of such manifolds, finishing a program initiated by Wall sixty years ago, with the exception of the $126$-dimensional case of the Kervaire invariant one problem. In particular, we find that the inertia group a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.00782v2-abstract-full').style.display = 'inline'; document.getElementById('2211.00782v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2211.00782v2-abstract-full" style="display: none;"> In this paper, we compute the inertia groups of $(n-1)$-connected, smooth, closed, oriented $2n$-manifolds where $n \geq 3$. As a consequence, we complete the diffeomorphism classification of such manifolds, finishing a program initiated by Wall sixty years ago, with the exception of the $126$-dimensional case of the Kervaire invariant one problem. In particular, we find that the inertia group always vanishes for $n \neq 4,8,9$ -- for $n \gg 0$, this was known by the work of several previous authors, including Wall, Stolz, and Burklund and Hahn with the first named author. When $n = 4,8,9$, we apply Kreck's modified surgery and a special case of Crowley's $Q$-form conjecture, proven by Nagy, to compute the inertia groups of these manifolds. In the cases $n=4,8$, our results recover unpublished work of Crowley--Nagy and Crowley--Olbermann. In contrast, we show that the homotopy and concordance inertia groups of $(n-1)$-connected, smooth, closed, oriented $2n$-manifolds with $n \geq 3$ always vanish. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.00782v2-abstract-full').style.display = 'none'; document.getElementById('2211.00782v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 November, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">v2: some improvements and minor corrections. Now 38 pages; comments still welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> CPH-GEOTOP-DNRF151 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2210.16406">arXiv:2210.16406</a> <span> [<a href="https://arxiv.org/pdf/2210.16406">pdf</a>, <a href="https://arxiv.org/format/2210.16406">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Gallai's Conjecture for Complete and "Nearly Complete" Graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+H">Hua Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Andrew Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2210.16406v1-abstract-short" style="display: inline;"> The famous Gallai's Conjecture states that any connected graph with n vertices has a path decomposition containing at most (n+1)/2 paths. In this note, we explore graphs generated from removing edges from complete graphs. We first provide an explicit construction for a path decomposition of complete graphs that satisfies Gallai's Conjecture. We then use that construction to prove that we can remov… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.16406v1-abstract-full').style.display = 'inline'; document.getElementById('2210.16406v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2210.16406v1-abstract-full" style="display: none;"> The famous Gallai's Conjecture states that any connected graph with n vertices has a path decomposition containing at most (n+1)/2 paths. In this note, we explore graphs generated from removing edges from complete graphs. We first provide an explicit construction for a path decomposition of complete graphs that satisfies Gallai's Conjecture. We then use that construction to prove that we can remove stars and certain tadpoles such that the resulting graph still satisfies Gallai's Conjecture. We also introduce a potential general approach through analyzing non-isomorphic path decompositions of complete graphs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.16406v1-abstract-full').style.display = 'none'; document.getElementById('2210.16406v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2210.01224">arXiv:2210.01224</a> <span> [<a href="https://arxiv.org/pdf/2210.01224">pdf</a>, <a href="https://arxiv.org/ps/2210.01224">ps</a>, <a href="https://arxiv.org/format/2210.01224">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> </div> </div> <p class="title is-5 mathjax"> On the factorization invariants of arithmetical congruence monoids </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chapman%2C+S+T">Scott T. Chapman</a>, <a href="/search/math?searchtype=author&query=Liu%2C+C">Caroline Liu</a>, <a href="/search/math?searchtype=author&query=Ma%2C+A">Annabel Ma</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A">Andrew Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2210.01224v3-abstract-short" style="display: inline;"> In this paper, we study various factorization invariants of arithmetical congruence monoids. The invariants we investigate are the catenary degree, a measure of the maximum distance between any two factorizations of the same element, the length density, which describes the distribution of the factorization lengths of an element, and the omega primality, which measures how far an element is from be… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.01224v3-abstract-full').style.display = 'inline'; document.getElementById('2210.01224v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2210.01224v3-abstract-full" style="display: none;"> In this paper, we study various factorization invariants of arithmetical congruence monoids. The invariants we investigate are the catenary degree, a measure of the maximum distance between any two factorizations of the same element, the length density, which describes the distribution of the factorization lengths of an element, and the omega primality, which measures how far an element is from being prime. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.01224v3-abstract-full').style.display = 'none'; document.getElementById('2210.01224v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 January, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.11215">arXiv:2209.11215</a> <span> [<a href="https://arxiv.org/pdf/2209.11215">pdf</a>, <a href="https://arxiv.org/ps/2209.11215">ps</a>, <a href="https://arxiv.org/format/2209.11215">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+S">Sitan Chen</a>, <a href="/search/math?searchtype=author&query=Chewi%2C+S">Sinho Chewi</a>, <a href="/search/math?searchtype=author&query=Li%2C+J">Jerry Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yuanzhi Li</a>, <a href="/search/math?searchtype=author&query=Salim%2C+A">Adil Salim</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2209.11215v3-abstract-short" style="display: inline;"> We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale real-world generative models such as DALL$\cdot$E 2. Our main result is that, assuming accurate score estimates, such SGMs can efficiently sample from essentially any realistic data distribution. In contrast to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.11215v3-abstract-full').style.display = 'inline'; document.getElementById('2209.11215v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.11215v3-abstract-full" style="display: none;"> We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale real-world generative models such as DALL$\cdot$E 2. Our main result is that, assuming accurate score estimates, such SGMs can efficiently sample from essentially any realistic data distribution. In contrast to prior works, our results (1) hold for an $L^2$-accurate score estimate (rather than $L^\infty$-accurate); (2) do not require restrictive functional inequality conditions that preclude substantial non-log-concavity; (3) scale polynomially in all relevant problem parameters; and (4) match state-of-the-art complexity guarantees for discretization of the Langevin diffusion, provided that the score error is sufficiently small. We view this as strong theoretical justification for the empirical success of SGMs. We also examine SGMs based on the critically damped Langevin diffusion (CLD). Contrary to conventional wisdom, we provide evidence that the use of the CLD does not reduce the complexity of SGMs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.11215v3-abstract-full').style.display = 'none'; document.getElementById('2209.11215v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.04962">arXiv:2209.04962</a> <span> [<a href="https://arxiv.org/pdf/2209.04962">pdf</a>, <a href="https://arxiv.org/format/2209.04962">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> Exact Minimax Optimality of Spectral Methods in Phase Synchronization and Orthogonal Group Synchronization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A+Y">Anderson Ye Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2209.04962v2-abstract-short" style="display: inline;"> We study the performance of the spectral method for the phase synchronization problem with additive Gaussian noises and incomplete data. The spectral method utilizes the leading eigenvector of the data matrix followed by a normalization step. We prove that it achieves the minimax lower bound of the problem with a matching leading constant under a squared $\ell_2$ loss. This shows that the spectral… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.04962v2-abstract-full').style.display = 'inline'; document.getElementById('2209.04962v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.04962v2-abstract-full" style="display: none;"> We study the performance of the spectral method for the phase synchronization problem with additive Gaussian noises and incomplete data. The spectral method utilizes the leading eigenvector of the data matrix followed by a normalization step. We prove that it achieves the minimax lower bound of the problem with a matching leading constant under a squared $\ell_2$ loss. This shows that the spectral method has the same performance as more sophisticated procedures including maximum likelihood estimation, generalized power method, and semidefinite programming, as long as consistent parameter estimation is possible. To establish our result, we first have a novel choice of the population eigenvector, which enables us to establish the exact recovery of the spectral method when there is no additive noise. We then develop a new perturbation analysis toolkit for the leading eigenvector and show it can be well-approximated by its first-order approximation with a small $\ell_2$ error. We further extend our analysis to establish the exact minimax optimality of the spectral method for the orthogonal group synchronization. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.04962v2-abstract-full').style.display = 'none'; document.getElementById('2209.04962v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.10293">arXiv:2208.10293</a> <span> [<a href="https://arxiv.org/pdf/2208.10293">pdf</a>, <a href="https://arxiv.org/ps/2208.10293">ps</a>, <a href="https://arxiv.org/format/2208.10293">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> </div> <p class="title is-5 mathjax"> Mod $p$ homology of unordered configuration spaces of surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+M">Matthew Chen</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+Y">Adela YiYu Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.10293v2-abstract-short" style="display: inline;"> We provide a short proof that the dimensions of the mod $p$ homology groups of the unordered configuration space $B_k(T)$ of $k$ points in a torus are the same as its Betti numbers for $p>2$ and $k\leq p$. Hence the integral homology has no $p$-power torsion. The same argument works for the punctured genus $g$ surface with $g>0$, thereby recovering a result of Brantner-Hahn-Knudsen via Lubin-Tate… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.10293v2-abstract-full').style.display = 'inline'; document.getElementById('2208.10293v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.10293v2-abstract-full" style="display: none;"> We provide a short proof that the dimensions of the mod $p$ homology groups of the unordered configuration space $B_k(T)$ of $k$ points in a torus are the same as its Betti numbers for $p>2$ and $k\leq p$. Hence the integral homology has no $p$-power torsion. The same argument works for the punctured genus $g$ surface with $g>0$, thereby recovering a result of Brantner-Hahn-Knudsen via Lubin-Tate theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.10293v2-abstract-full').style.display = 'none'; document.getElementById('2208.10293v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.06046">arXiv:2208.06046</a> <span> [<a href="https://arxiv.org/pdf/2208.06046">pdf</a>, <a href="https://arxiv.org/format/2208.06046">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Finance">q-fin.MF</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Portfolio Management">q-fin.PM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Pricing of Securities">q-fin.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Trading and Market Microstructure">q-fin.TR</span> </div> </div> <p class="title is-5 mathjax"> Automated Market Making and Loss-Versus-Rebalancing </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Milionis%2C+J">Jason Milionis</a>, <a href="/search/math?searchtype=author&query=Moallemi%2C+C+C">Ciamac C. Moallemi</a>, <a href="/search/math?searchtype=author&query=Roughgarden%2C+T">Tim Roughgarden</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+L">Anthony Lee Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.06046v5-abstract-short" style="display: inline;"> We consider the market microstructure of automated market makers (AMMs) from the perspective of liquidity providers (LPs). Our central contribution is a ``Black-Scholes formula for AMMs''. We identify the main adverse selection cost incurred by LPs, which we call ``loss-versus-rebalancing'' (LVR, pronounced ``lever''). LVR captures costs incurred by AMM LPs due to stale prices that are picked off… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.06046v5-abstract-full').style.display = 'inline'; document.getElementById('2208.06046v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.06046v5-abstract-full" style="display: none;"> We consider the market microstructure of automated market makers (AMMs) from the perspective of liquidity providers (LPs). Our central contribution is a ``Black-Scholes formula for AMMs''. We identify the main adverse selection cost incurred by LPs, which we call ``loss-versus-rebalancing'' (LVR, pronounced ``lever''). LVR captures costs incurred by AMM LPs due to stale prices that are picked off by better informed arbitrageurs. We derive closed-form expressions for LVR applicable to all automated market makers. Our model is quantitatively realistic, matching actual LP returns empirically, and shows how CFMM protocols can be redesigned to reduce or eliminate LVR. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.06046v5-abstract-full').style.display = 'none'; document.getElementById('2208.06046v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">63 pages, 7 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.00055">arXiv:2208.00055</a> <span> [<a href="https://arxiv.org/pdf/2208.00055">pdf</a>, <a href="https://arxiv.org/format/2208.00055">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jfa.2023.109883">10.1016/j.jfa.2023.109883 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Periodic approximations in inverse spectral problems for canonical Hamiltonian systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Poltoratski%2C+A">Alexei Poltoratski</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Ashley Ran Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.00055v1-abstract-short" style="display: inline;"> This note is devoted to inverse spectral problems for canonical Hamiltonian systems on the half-line. An approach to inverse spectral problems based on the use of truncated Toeplitz operators has been especially effective in the case when the spectral measure of the system is a locally finite periodic measure (see \cite{MP}). In this note we extend the periodic algorithm to the case of non-periodi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.00055v1-abstract-full').style.display = 'inline'; document.getElementById('2208.00055v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.00055v1-abstract-full" style="display: none;"> This note is devoted to inverse spectral problems for canonical Hamiltonian systems on the half-line. An approach to inverse spectral problems based on the use of truncated Toeplitz operators has been especially effective in the case when the spectral measure of the system is a locally finite periodic measure (see \cite{MP}). In this note we extend the periodic algorithm to the case of non-periodic measures by considering periodizations of a spectral measure and showing that the Hamiltonians corresponding to the periodizations converge to the Hamiltonian of the original measure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.00055v1-abstract-full').style.display = 'none'; document.getElementById('2208.00055v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2207.10665">arXiv:2207.10665</a> <span> [<a href="https://arxiv.org/pdf/2207.10665">pdf</a>, <a href="https://arxiv.org/format/2207.10665">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> One-dimensional Tensor Network Recovery </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+Z">Ziang Chen</a>, <a href="/search/math?searchtype=author&query=Lu%2C+J">Jianfeng Lu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2207.10665v3-abstract-short" style="display: inline;"> We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is $O(d\log d)$ for $d$-th order tensors. We prove that our algorithms can almost surely recover the correct graph or permutation when tensor entries can be observed without noise. We furt… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.10665v3-abstract-full').style.display = 'inline'; document.getElementById('2207.10665v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2207.10665v3-abstract-full" style="display: none;"> We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is $O(d\log d)$ for $d$-th order tensors. We prove that our algorithms can almost surely recover the correct graph or permutation when tensor entries can be observed without noise. We further establish the robustness of our algorithms against observational noise. The theoretical results are validated by numerical experiments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.10665v3-abstract-full').style.display = 'none'; document.getElementById('2207.10665v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2206.08756">arXiv:2206.08756</a> <span> [<a href="https://arxiv.org/pdf/2206.08756">pdf</a>, <a href="https://arxiv.org/format/2206.08756">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yuetian Luo</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+A+R">Anru R. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2206.08756v3-abstract-short" style="display: inline;"> We study the tensor-on-tensor regression, where the goal is to connect tensor responses to tensor covariates with a low Tucker rank parameter tensor/matrix without the prior knowledge of its intrinsic rank. We propose the Riemannian gradient descent (RGD) and Riemannian Gauss-Newton (RGN) methods and cope with the challenge of unknown rank by studying the effect of rank over-parameterization. We p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.08756v3-abstract-full').style.display = 'inline'; document.getElementById('2206.08756v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2206.08756v3-abstract-full" style="display: none;"> We study the tensor-on-tensor regression, where the goal is to connect tensor responses to tensor covariates with a low Tucker rank parameter tensor/matrix without the prior knowledge of its intrinsic rank. We propose the Riemannian gradient descent (RGD) and Riemannian Gauss-Newton (RGN) methods and cope with the challenge of unknown rank by studying the effect of rank over-parameterization. We provide the first convergence guarantee for the general tensor-on-tensor regression by showing that RGD and RGN respectively converge linearly and quadratically to a statistically optimal estimate in both rank correctly-parameterized and over-parameterized settings. Our theory reveals an intriguing phenomenon: Riemannian optimization methods naturally adapt to over-parameterization without modifications to their implementation. We also prove the statistical-computational gap in scalar-on-tensor regression by a direct low-degree polynomial argument. Our theory demonstrates a "blessing of statistical-computational gap" phenomenon: in a wide range of scenarios in tensor-on-tensor regression for tensors of order three or higher, the computationally required sample size matches what is needed by moderate rank over-parameterization when considering computationally feasible estimators, while there are no such benefits in the matrix settings. This shows moderate rank over-parameterization is essentially "cost-free" in terms of sample size in tensor-on-tensor regression of order three or higher. Finally, we conduct simulation studies to show the advantages of our proposed methods and to corroborate our theoretical findings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.08756v3-abstract-full').style.display = 'none'; document.getElementById('2206.08756v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 June, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.14855">arXiv:2205.14855</a> <span> [<a href="https://arxiv.org/pdf/2205.14855">pdf</a>, <a href="https://arxiv.org/ps/2205.14855">ps</a>, <a href="https://arxiv.org/format/2205.14855">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> Leave-one-out Singular Subspace Perturbation Analysis for Spectral Clustering </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A+Y">Anderson Y. Zhang</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+H+H">Harrison H. Zhou</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2205.14855v2-abstract-short" style="display: inline;"> The singular subspaces perturbation theory is of fundamental importance in probability and statistics. It has various applications across different fields. We consider two arbitrary matrices where one is a leave-one-column-out submatrix of the other one and establish a novel perturbation upper bound for the distance between the two corresponding singular subspaces. It is well-suited for mixture mo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.14855v2-abstract-full').style.display = 'inline'; document.getElementById('2205.14855v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.14855v2-abstract-full" style="display: none;"> The singular subspaces perturbation theory is of fundamental importance in probability and statistics. It has various applications across different fields. We consider two arbitrary matrices where one is a leave-one-column-out submatrix of the other one and establish a novel perturbation upper bound for the distance between the two corresponding singular subspaces. It is well-suited for mixture models and results in a sharper and finer statistical analysis than classical perturbation bounds such as Wedin's Theorem. Empowered by this leave-one-out perturbation theory, we provide a deterministic entrywise analysis for the performance of spectral clustering under mixture models. Our analysis leads to an explicit exponential error rate for spectral clustering of sub-Gaussian mixture models. For the mixture of isotropic Gaussians, the rate is optimal under a weaker signal-to-noise condition than that of L{枚}ffler et al. (2021). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.14855v2-abstract-full').style.display = 'none'; document.getElementById('2205.14855v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2022. </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Zhang%2C+A&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Zhang%2C+A&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 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