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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=Han%2C+R&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Han%2C+R&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Han%2C+R&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </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/2408.15962">arXiv:2408.15962</a> <span> [<a href="https://arxiv.org/pdf/2408.15962">pdf</a>, <a href="https://arxiv.org/ps/2408.15962">ps</a>, <a href="https://arxiv.org/format/2408.15962">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</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"> H枚lder continuity of the integrated density of states for Liouville frequencies </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Schlag%2C+W">Wilhelm Schlag</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="2408.15962v1-abstract-short" style="display: inline;"> We prove H枚lder continuity of the Lyapunov exponent $L(蠅,E)$ and the integrated density of states at energies that satisfy $L(蠅,E)>4魏(蠅,E)\cdot 尾(蠅)\geq 0$ for general analytic potentials, with $魏(蠅,E)$ being Avila's acceleration. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.15962v1-abstract-full" style="display: none;"> We prove H枚lder continuity of the Lyapunov exponent $L(蠅,E)$ and the integrated density of states at energies that satisfy $L(蠅,E)>4魏(蠅,E)\cdot 尾(蠅)\geq 0$ for general analytic potentials, with $魏(蠅,E)$ being Avila's acceleration. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.15962v1-abstract-full').style.display = 'none'; document.getElementById('2408.15962v1-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 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/2405.13153">arXiv:2405.13153</a> <span> [<a href="https://arxiv.org/pdf/2405.13153">pdf</a>, <a href="https://arxiv.org/format/2405.13153">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">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Max-sliced Wasserstein concentration and uniform ratio bounds of empirical measures on RKHS </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Ruiyu Han</a>, <a href="/search/math?searchtype=author&query=Rush%2C+C">Cynthia Rush</a>, <a href="/search/math?searchtype=author&query=Wiesel%2C+J">Johannes Wiesel</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="2405.13153v1-abstract-short" style="display: inline;"> Optimal transport and the Wasserstein distance $\mathcal{W}_p$ have recently seen a number of applications in the fields of statistics, machine learning, data science, and the physical sciences. These applications are however severely restricted by the curse of dimensionality, meaning that the number of data points needed to estimate these problems accurately increases exponentially in the dimensi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.13153v1-abstract-full').style.display = 'inline'; document.getElementById('2405.13153v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.13153v1-abstract-full" style="display: none;"> Optimal transport and the Wasserstein distance $\mathcal{W}_p$ have recently seen a number of applications in the fields of statistics, machine learning, data science, and the physical sciences. These applications are however severely restricted by the curse of dimensionality, meaning that the number of data points needed to estimate these problems accurately increases exponentially in the dimension. To alleviate this problem, a number of variants of $\mathcal{W}_p$ have been introduced. We focus here on one of these variants, namely the max-sliced Wasserstein metric $\overline{\mathcal{W}}_p$. This metric reduces the high-dimensional minimization problem given by $\mathcal{W}_p$ to a maximum of one-dimensional measurements in an effort to overcome the curse of dimensionality. In this note we derive concentration results and upper bounds on the expectation of $\overline{\mathcal{W}}_p$ between the true and empirical measure on unbounded reproducing kernel Hilbert spaces. We show that, under quite generic assumptions, probability measures concentrate uniformly fast in one-dimensional subspaces, at (nearly) parametric rates. Our results rely on an improvement of currently known bounds for $\overline{\mathcal{W}}_p$ in the finite-dimensional case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.13153v1-abstract-full').style.display = 'none'; document.getElementById('2405.13153v1-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 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.07810">arXiv:2405.07810</a> <span> [<a href="https://arxiv.org/pdf/2405.07810">pdf</a>, <a href="https://arxiv.org/ps/2405.07810">ps</a>, <a href="https://arxiv.org/format/2405.07810">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> <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="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Sharp localization on the first supercritical stratum for Liouville frequencies </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</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="2405.07810v1-abstract-short" style="display: inline;"> We establish Anderson localization for Schr枚dinger operators with even analytic potentials on the first supercritical stratum for Liouville frequencies in the sharp regime $\{E: L(蠅,E)>尾(蠅)>0, 魏(蠅,E)=1\}$, with $魏(蠅,E)$ being Avila's acceleration. This paper builds on the large deviation measure estimate and complexity bound scheme, originally developed for Diophantine frequencies by Bourgain, Gol… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.07810v1-abstract-full').style.display = 'inline'; document.getElementById('2405.07810v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.07810v1-abstract-full" style="display: none;"> We establish Anderson localization for Schr枚dinger operators with even analytic potentials on the first supercritical stratum for Liouville frequencies in the sharp regime $\{E: L(蠅,E)>尾(蠅)>0, 魏(蠅,E)=1\}$, with $魏(蠅,E)$ being Avila's acceleration. This paper builds on the large deviation measure estimate and complexity bound scheme, originally developed for Diophantine frequencies by Bourgain, Goldstein and Schlag \cites{BG,BGS1,BGS2}, and the improved complexity bounds in \cite{HS1}. Additionally, it strengthens the large deviation estimates for weak Liouville frequencies in \cite{HZ}. We also introduce new ideas to handle Liouville frequencies in a sharp way. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.07810v1-abstract-full').style.display = 'none'; document.getElementById('2405.07810v1-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 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.08463">arXiv:2401.08463</a> <span> [<a href="https://arxiv.org/pdf/2401.08463">pdf</a>, <a href="https://arxiv.org/format/2401.08463">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">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Statistical inference for pairwise comparison models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Ruijian Han</a>, <a href="/search/math?searchtype=author&query=Tang%2C+W">Wenlu Tang</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Y">Yiming Xu</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="2401.08463v2-abstract-short" style="display: inline;"> Pairwise comparison models have been widely used for utility evaluation and ranking across various fields. The increasing scale of problems today underscores the need to understand statistical inference in these models when the number of subjects diverges, a topic currently lacking in the literature except in a few special instances. To partially address this gap, this paper establishes a near-opt… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.08463v2-abstract-full').style.display = 'inline'; document.getElementById('2401.08463v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.08463v2-abstract-full" style="display: none;"> Pairwise comparison models have been widely used for utility evaluation and ranking across various fields. The increasing scale of problems today underscores the need to understand statistical inference in these models when the number of subjects diverges, a topic currently lacking in the literature except in a few special instances. To partially address this gap, this paper establishes a near-optimal asymptotic normality result for the maximum likelihood estimator in a broad class of pairwise comparison models, as well as a non-asymptotic convergence rate for each individual subject under comparison. The key idea lies in identifying the Fisher information matrix as a weighted graph Laplacian, which can be studied via a meticulous spectral analysis. Our findings provide a unified theory for performing statistical inference in a wide range of pairwise comparison models beyond the Bradley--Terry model, benefiting practitioners with theoretical guarantees for their use. Simulations utilizing synthetic data are conducted to validate the asymptotic normality result, followed by a hypothesis test using a tennis competition dataset. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.08463v2-abstract-full').style.display = 'none'; document.getElementById('2401.08463v2-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 16 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">28 pages; include additional results</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.03423">arXiv:2309.03423</a> <span> [<a href="https://arxiv.org/pdf/2309.03423">pdf</a>, <a href="https://arxiv.org/ps/2309.03423">ps</a>, <a href="https://arxiv.org/format/2309.03423">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> </div> <p class="title is-5 mathjax"> Non-perturbative localization for quasi-periodic Jacobi block matrices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Schlag%2C+W">Wilhelm Schlag</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="2309.03423v1-abstract-short" style="display: inline;"> We prove non-perturbative Anderson localization for quasi-periodic Jacobi block matrix operators assuming non-vanishing of all Lyapunov exponents. The base dynamics on tori $\mathbb{T}^b$ is assumed to be a Diophantine rotation. Results on arithmetic localization are obtained for $b=1$, and applications to the skew shift, stacked graphene, XY spin chains, and coupled Harper models are discussed. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.03423v1-abstract-full" style="display: none;"> We prove non-perturbative Anderson localization for quasi-periodic Jacobi block matrix operators assuming non-vanishing of all Lyapunov exponents. The base dynamics on tori $\mathbb{T}^b$ is assumed to be a Diophantine rotation. Results on arithmetic localization are obtained for $b=1$, and applications to the skew shift, stacked graphene, XY spin chains, and coupled Harper models are discussed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.03423v1-abstract-full').style.display = 'none'; document.getElementById('2309.03423v1-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, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 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">56 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/2307.11987">arXiv:2307.11987</a> <span> [<a href="https://arxiv.org/pdf/2307.11987">pdf</a>, <a href="https://arxiv.org/format/2307.11987">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> </div> </div> <p class="title is-5 mathjax"> A Monotone Discretization for the Fractional Obstacle Problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rubing Han</a>, <a href="/search/math?searchtype=author&query=Wu%2C+S">Shuonan Wu</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+H">Hao 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="2307.11987v2-abstract-short" style="display: inline;"> We introduce a novel monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. This problem is prevalent in mathematical finance, particle systems, and elastic theory. By leveraging insights from the successful monotone discretization of the fractional Laplacian, we esta… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.11987v2-abstract-full').style.display = 'inline'; document.getElementById('2307.11987v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.11987v2-abstract-full" style="display: none;"> We introduce a novel monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. This problem is prevalent in mathematical finance, particle systems, and elastic theory. By leveraging insights from the successful monotone discretization of the fractional Laplacian, we establish uniform boundedness, solution existence, and uniqueness for the numerical solutions of the fractional obstacle problem. We employ a policy iteration approach for efficient solution of discrete nonlinear problems and prove its finite convergence. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the method's efficacy. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.11987v2-abstract-full').style.display = 'none'; document.getElementById('2307.11987v2-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 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 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, 7 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R11; 65N06; 65N12; 65N15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.05802">arXiv:2307.05802</a> <span> [<a href="https://arxiv.org/pdf/2307.05802">pdf</a>, <a href="https://arxiv.org/ps/2307.05802">ps</a>, <a href="https://arxiv.org/format/2307.05802">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</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="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Sliced Wasserstein Distance between Probability Measures on Hilbert Spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Ruiyu Han</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="2307.05802v2-abstract-short" style="display: inline;"> The sliced Wasserstein distance as well as its variants have been widely considered in comparing probability measures defined on $\mathbb R^d$. Here we derive the notion of sliced Wasserstein distance for measures on an infinite dimensional separable Hilbert spaces, depict the relation between sliced Wasserstein distance and narrow convergence of measures and quantize the approximation via empiric… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.05802v2-abstract-full').style.display = 'inline'; document.getElementById('2307.05802v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.05802v2-abstract-full" style="display: none;"> The sliced Wasserstein distance as well as its variants have been widely considered in comparing probability measures defined on $\mathbb R^d$. Here we derive the notion of sliced Wasserstein distance for measures on an infinite dimensional separable Hilbert spaces, depict the relation between sliced Wasserstein distance and narrow convergence of measures and quantize the approximation via empirical measures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.05802v2-abstract-full').style.display = 'none'; document.getElementById('2307.05802v2-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 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 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">11 pages, 0 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53B12 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2306.15122">arXiv:2306.15122</a> <span> [<a href="https://arxiv.org/pdf/2306.15122">pdf</a>, <a href="https://arxiv.org/ps/2306.15122">ps</a>, <a href="https://arxiv.org/format/2306.15122">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</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"> Non-perturbative localization on the strip and Avila's almost reducibility conjecture </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Schlag%2C+W">Wilhelm Schlag</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="2306.15122v2-abstract-short" style="display: inline;"> We prove non-perturbative Anderson localization and almost localization for a family of quasi-periodic operators on the strip. As an application we establish Avila's almost reducibility conjecture for Schr枚dinger operators with trigonometric potentials and all Diophantine frequencies, whose proof for analytic potentials was announced in Avila's 2015 Acta paper. As part of our analysis, we derive a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.15122v2-abstract-full').style.display = 'inline'; document.getElementById('2306.15122v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2306.15122v2-abstract-full" style="display: none;"> We prove non-perturbative Anderson localization and almost localization for a family of quasi-periodic operators on the strip. As an application we establish Avila's almost reducibility conjecture for Schr枚dinger operators with trigonometric potentials and all Diophantine frequencies, whose proof for analytic potentials was announced in Avila's 2015 Acta paper. As part of our analysis, we derive a non-selfadjoint version of Haro and Puig's formula connecting Lyapunov exponents of the dual model to those of the original operator. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.15122v2-abstract-full').style.display = 'none'; document.getElementById('2306.15122v2-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 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 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">67 pages. Comments 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/2306.02821">arXiv:2306.02821</a> <span> [<a href="https://arxiv.org/pdf/2306.02821">pdf</a>, <a href="https://arxiv.org/format/2306.02821">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"> A unified analysis of likelihood-based estimators in the Plackett--Luce model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Ruijian Han</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Y">Yiming Xu</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="2306.02821v3-abstract-short" style="display: inline;"> The Plackett--Luce model has been extensively used for rank aggregation in social choice theory. A central question in this model concerns estimating the utility vector that governs the model's likelihood. In this paper, we investigate the asymptotic theory of utility vector estimation by maximizing different types of likelihood, such as full, marginal, and quasi-likelihood. Starting from interpre… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.02821v3-abstract-full').style.display = 'inline'; document.getElementById('2306.02821v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2306.02821v3-abstract-full" style="display: none;"> The Plackett--Luce model has been extensively used for rank aggregation in social choice theory. A central question in this model concerns estimating the utility vector that governs the model's likelihood. In this paper, we investigate the asymptotic theory of utility vector estimation by maximizing different types of likelihood, such as full, marginal, and quasi-likelihood. Starting from interpreting the estimating equations of these estimators to gain some initial insights, we analyze their asymptotic behavior as the number of compared objects increases. In particular, we establish both the uniform consistency and asymptotic normality of these estimators and discuss the trade-off between statistical efficiency and computational complexity. For generality, our results are proven for deterministic graph sequences under appropriate graph topology conditions. These conditions are shown to be revealing and sharp when applied to common sampling scenarios, such as nonuniform random hypergraph models and hypergraph stochastic block models. Numerical results are provided to support our findings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.02821v3-abstract-full').style.display = 'none'; document.getElementById('2306.02821v3-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 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">58 pages; extended asymptotic normality results to deterministic graph sequences and fixed typos</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2304.11538">arXiv:2304.11538</a> <span> [<a href="https://arxiv.org/pdf/2304.11538">pdf</a>, <a href="https://arxiv.org/format/2304.11538">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</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"> HV Geometry for Signal Comparison </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Ruiyu Han</a>, <a href="/search/math?searchtype=author&query=Slep%C4%8Dev%2C+D">Dejan Slep膷ev</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yunan Yang</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="2304.11538v1-abstract-short" style="display: inline;"> In order to compare and interpolate signals, we investigate a Riemannian geometry on the space of signals. The metric allows discontinuous signals and measures both horizontal (thus providing many benefits of the Wasserstein metric) and vertical deformations. Moreover, it allows for signed signals, which overcomes the main deficiency of optimal transportation-based metrics in signal processing. We… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.11538v1-abstract-full').style.display = 'inline'; document.getElementById('2304.11538v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2304.11538v1-abstract-full" style="display: none;"> In order to compare and interpolate signals, we investigate a Riemannian geometry on the space of signals. The metric allows discontinuous signals and measures both horizontal (thus providing many benefits of the Wasserstein metric) and vertical deformations. Moreover, it allows for signed signals, which overcomes the main deficiency of optimal transportation-based metrics in signal processing. We characterize the metric properties of the space of signals and establish the regularity and stability of geodesics. Furthermore, we introduce an efficient numerical scheme to compute the geodesics and present several experiments which highlight the nature of the metric. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.11538v1-abstract-full').style.display = 'none'; document.getElementById('2304.11538v1-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 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 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">34 pages, 8 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/2303.06422">arXiv:2303.06422</a> <span> [<a href="https://arxiv.org/pdf/2303.06422">pdf</a>, <a href="https://arxiv.org/format/2303.06422">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</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="Applications">stat.AP</span> </div> </div> <p class="title is-5 mathjax"> An approximate control variates approach to multifidelity distribution estimation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Ruijian Han</a>, <a href="/search/math?searchtype=author&query=Kramer%2C+B">Boris Kramer</a>, <a href="/search/math?searchtype=author&query=Lee%2C+D">Dongjin Lee</a>, <a href="/search/math?searchtype=author&query=Narayan%2C+A">Akil Narayan</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Y">Yiming Xu</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="2303.06422v2-abstract-short" style="display: inline;"> Forward simulation-based uncertainty quantification that studies the distribution of quantities of interest (QoI) is a crucial component for computationally robust engineering design and prediction. There is a large body of literature devoted to accurately assessing statistics of QoIs, and in particular, multilevel or multifidelity approaches are known to be effective, leveraging cost-accuracy tra… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.06422v2-abstract-full').style.display = 'inline'; document.getElementById('2303.06422v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.06422v2-abstract-full" style="display: none;"> Forward simulation-based uncertainty quantification that studies the distribution of quantities of interest (QoI) is a crucial component for computationally robust engineering design and prediction. There is a large body of literature devoted to accurately assessing statistics of QoIs, and in particular, multilevel or multifidelity approaches are known to be effective, leveraging cost-accuracy tradeoffs between a given ensemble of models. However, effective algorithms that can estimate the full distribution of QoIs are still under active development. In this paper, we introduce a general multifidelity framework for estimating the cumulative distribution function (CDF) of a vector-valued QoI associated with a high-fidelity model under a budget constraint. Given a family of appropriate control variates obtained from lower-fidelity surrogates, our framework involves identifying the most cost-effective model subset and then using it to build an approximate control variates estimator for the target CDF. We instantiate the framework by constructing a family of control variates using intermediate linear approximators and rigorously analyze the corresponding algorithm. Our analysis reveals that the resulting CDF estimator is uniformly consistent and asymptotically optimal as the budget tends to infinity, with only mild moment and regularity assumptions on the joint distribution of QoIs. The approach provides a robust multifidelity CDF estimator that is adaptive to the available budget, does not require \textit{a priori} knowledge of cross-model statistics or model hierarchy, and applies to multiple dimensions. We demonstrate the efficiency and robustness of the approach using test examples of parametric PDEs and stochastic differential equations including both academic instances and more challenging engineering problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.06422v2-abstract-full').style.display = 'none'; document.getElementById('2303.06422v2-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 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">41 pages, added additional numerical experiments</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.05988">arXiv:2212.05988</a> <span> [<a href="https://arxiv.org/pdf/2212.05988">pdf</a>, <a href="https://arxiv.org/ps/2212.05988">ps</a>, <a href="https://arxiv.org/format/2212.05988">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</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"> Avila's acceleration via zeros of determinants, and applications to Schr枚dinger cocycles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Schlag%2C+W">Wilhelm Schlag</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="2212.05988v2-abstract-short" style="display: inline;"> In this paper we give a characterization of Avila's quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove $尾$-H枚lder continuity of the integrated density of states for supercritical quasi-periodic Schr枚dinger operators restricted to the $\ell$-th stratum, for any $尾<(2(\ell-1))^{-1}$ and $\ell\ge2$. We esta… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.05988v2-abstract-full').style.display = 'inline'; document.getElementById('2212.05988v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.05988v2-abstract-full" style="display: none;"> In this paper we give a characterization of Avila's quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove $尾$-H枚lder continuity of the integrated density of states for supercritical quasi-periodic Schr枚dinger operators restricted to the $\ell$-th stratum, for any $尾<(2(\ell-1))^{-1}$ and $\ell\ge2$. We establish Anderson localization for all Diophantine frequencies for the operator with even analytic potential function on the first supercritical stratum, which has positive measure if it is nonempty. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.05988v2-abstract-full').style.display = 'none'; document.getElementById('2212.05988v2-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 January, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 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">22 pages. Comments 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/2205.10151">arXiv:2205.10151</a> <span> [<a href="https://arxiv.org/pdf/2205.10151">pdf</a>, <a href="https://arxiv.org/ps/2205.10151">ps</a>, <a href="https://arxiv.org/format/2205.10151">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"> Discussion of "Vintage factor analysis with Varimax performs statistical inference" </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rungang Han</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="2205.10151v1-abstract-short" style="display: inline;"> We wholeheartedly congratulate Drs. Rohe and Zeng for their insightful paper \cite{rohe2020vintage} on vintage factor analysis with Varimax rotation. This note discusses the conditions to guarantee Varimax consistently recovers the subspace rotation. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.10151v1-abstract-full" style="display: none;"> We wholeheartedly congratulate Drs. Rohe and Zeng for their insightful paper \cite{rohe2020vintage} on vintage factor analysis with Varimax rotation. This note discusses the conditions to guarantee Varimax consistently recovers the subspace rotation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.10151v1-abstract-full').style.display = 'none'; document.getElementById('2205.10151v1-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 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.04021">arXiv:2205.04021</a> <span> [<a href="https://arxiv.org/pdf/2205.04021">pdf</a>, <a href="https://arxiv.org/ps/2205.04021">ps</a>, <a href="https://arxiv.org/format/2205.04021">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Anti-resonances and sharp analysis of Maryland localization for all parameters </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Jitomirskaya%2C+S">Svetlana Jitomirskaya</a>, <a href="/search/math?searchtype=author&query=Yang%2C+F">Fan Yang</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="2205.04021v1-abstract-short" style="display: inline;"> We develop the technique to prove localization through the analysis of eigenfunctions in presence of both exponential frequency resonances and exponential phase barriers (anti-resonances) and use it to prove localization for the Maryland model for all parameters. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.04021v1-abstract-full" style="display: none;"> We develop the technique to prove localization through the analysis of eigenfunctions in presence of both exponential frequency resonances and exponential phase barriers (anti-resonances) and use it to prove localization for the Maryland model for all parameters. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.04021v1-abstract-full').style.display = 'none'; document.getElementById('2205.04021v1-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, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">45 pages. Comments 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/2109.09308">arXiv:2109.09308</a> <span> [<a href="https://arxiv.org/pdf/2109.09308">pdf</a>, <a href="https://arxiv.org/ps/2109.09308">ps</a>, <a href="https://arxiv.org/format/2109.09308">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> </div> </div> <p class="title is-5 mathjax"> A monotone discretization for integral fractional Laplacian on bounded Lipschitz domains: Pointwise error estimates under H枚lder regularity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rubing Han</a>, <a href="/search/math?searchtype=author&query=Wu%2C+S">Shuonan Wu</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="2109.09308v3-abstract-short" style="display: inline;"> We propose a monotone discretization for the integral fractional Laplace equation on bounded Lipschitz domains with the homogeneous Dirichlet boundary condition. The method is inspired by a quadrature-based finite difference method of Huang and Oberman, but is defined on unstructured grids in arbitrary dimensions with a more flexible domain for approximating singular integral. The scale of the sin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.09308v3-abstract-full').style.display = 'inline'; document.getElementById('2109.09308v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2109.09308v3-abstract-full" style="display: none;"> We propose a monotone discretization for the integral fractional Laplace equation on bounded Lipschitz domains with the homogeneous Dirichlet boundary condition. The method is inspired by a quadrature-based finite difference method of Huang and Oberman, but is defined on unstructured grids in arbitrary dimensions with a more flexible domain for approximating singular integral. The scale of the singular integral domain not only depends on the local grid size, but also on the distance to the boundary, since the H枚lder coefficient of the solution deteriorates as it approaches the boundary. By using a discrete barrier function that also reflects the distance to the boundary, we show optimal pointwise convergence rates in terms of the H枚lder regularity of the data on both quasi-uniform and graded grids. Several numerical examples are provided to illustrate the sharpness of the theoretical results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.09308v3-abstract-full').style.display = 'none'; document.getElementById('2109.09308v3-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, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R11; 65N06; 65N12; 65N15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2108.05767">arXiv:2108.05767</a> <span> [<a href="https://arxiv.org/pdf/2108.05767">pdf</a>, <a href="https://arxiv.org/format/2108.05767">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</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="Machine Learning">stat.ML</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/10.1093/imaiai/iaac008">10.1093/imaiai/iaac008 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Probabilistic methods for approximate archetypal analysis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Ruijian Han</a>, <a href="/search/math?searchtype=author&query=Osting%2C+B">Braxton Osting</a>, <a href="/search/math?searchtype=author&query=Wang%2C+D">Dong Wang</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Y">Yiming Xu</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="2108.05767v3-abstract-short" style="display: inline;"> Archetypal analysis is an unsupervised learning method for exploratory data analysis. One major challenge that limits the applicability of archetypal analysis in practice is the inherent computational complexity of the existing algorithms. In this paper, we provide a novel approximation approach to partially address this issue. Utilizing probabilistic ideas from high-dimensional geometry, we intro… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.05767v3-abstract-full').style.display = 'inline'; document.getElementById('2108.05767v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2108.05767v3-abstract-full" style="display: none;"> Archetypal analysis is an unsupervised learning method for exploratory data analysis. One major challenge that limits the applicability of archetypal analysis in practice is the inherent computational complexity of the existing algorithms. In this paper, we provide a novel approximation approach to partially address this issue. Utilizing probabilistic ideas from high-dimensional geometry, we introduce two preprocessing techniques to reduce the dimension and representation cardinality of the data, respectively. We prove that provided the data is approximately embedded in a low-dimensional linear subspace and the convex hull of the corresponding representations is well approximated by a polytope with a few vertices, our method can effectively reduce the scaling of archetypal analysis. Moreover, the solution of the reduced problem is near-optimal in terms of prediction errors. Our approach can be combined with other acceleration techniques to further mitigate the intrinsic complexity of archetypal analysis. We demonstrate the usefulness of our results by applying our method to summarize several moderately large-scale datasets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.05767v3-abstract-full').style.display = 'none'; document.getElementById('2108.05767v3-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 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2021. </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">Information and Inference: A Journal of the IMA, 2022</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2108.04437">arXiv:2108.04437</a> <span> [<a href="https://arxiv.org/pdf/2108.04437">pdf</a>, <a href="https://arxiv.org/format/2108.04437">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"> Statistical Inference in High-dimensional Generalized Linear Models with Streaming Data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Lan Luo</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Ruijian Han</a>, <a href="/search/math?searchtype=author&query=Lin%2C+Y">Yuanyuan Lin</a>, <a href="/search/math?searchtype=author&query=Huang%2C+J">Jian Huang</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="2108.04437v1-abstract-short" style="display: inline;"> In this paper we develop an online statistical inference approach for high-dimensional generalized linear models with streaming data for real-time estimation and inference. We propose an online debiased lasso (ODL) method to accommodate the special structure of streaming data. ODL differs from offline debiased lasso in two important aspects. First, in computing the estimate at the current stage, i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.04437v1-abstract-full').style.display = 'inline'; document.getElementById('2108.04437v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2108.04437v1-abstract-full" style="display: none;"> In this paper we develop an online statistical inference approach for high-dimensional generalized linear models with streaming data for real-time estimation and inference. We propose an online debiased lasso (ODL) method to accommodate the special structure of streaming data. ODL differs from offline debiased lasso in two important aspects. First, in computing the estimate at the current stage, it only uses summary statistics of the historical data. Second, in addition to debiasing an online lasso estimator, ODL corrects an approximation error term arising from nonlinear online updating with streaming data. We show that the proposed online debiased estimators for the GLMs are consistent and asymptotically normal. This result provides a theoretical basis for carrying out real-time interim statistical inference with streaming data. Extensive numerical experiments are conducted to evaluate the performance of the proposed ODL method. These experiments demonstrate the effectiveness of our algorithm and support the theoretical results. A streaming dataset from the National Automotive Sampling System-Crashworthiness Data System is analyzed to illustrate the application of the proposed method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.04437v1-abstract-full').style.display = 'none'; document.getElementById('2108.04437v1-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, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2021. </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">Lan Luo and Ruijian Han contributed equally to this work. Co-corresponding authors: Yuanyuan Lin (email: ylin@sta.cuhk.edu.hk) and Jian Huang (email: jian-huang@uiowa.edu)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2108.04201">arXiv:2108.04201</a> <span> [<a href="https://arxiv.org/pdf/2108.04201">pdf</a>, <a href="https://arxiv.org/format/2108.04201">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> </div> </div> <p class="title is-5 mathjax"> Guaranteed Functional Tensor Singular Value Decomposition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rungang Han</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="2108.04201v3-abstract-short" style="display: inline;"> This paper introduces the functional tensor singular value decomposition (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. The problem is motivated by high-order longitudinal data analysis. Our model assumes the observed data to be a random realization of an approximate CP low-rank functional tensor measured on a discrete time grid. Inco… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.04201v3-abstract-full').style.display = 'inline'; document.getElementById('2108.04201v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2108.04201v3-abstract-full" style="display: none;"> This paper introduces the functional tensor singular value decomposition (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. The problem is motivated by high-order longitudinal data analysis. Our model assumes the observed data to be a random realization of an approximate CP low-rank functional tensor measured on a discrete time grid. Incorporating tensor algebra and the theory of Reproducing Kernel Hilbert Space (RKHS), we propose a novel RKHS-based constrained power iteration with spectral initialization. Our method can successfully estimate both singular vectors and functions of the low-rank structure in the observed data. With mild assumptions, we establish the non-asymptotic contractive error bounds for the proposed algorithm. The superiority of the proposed framework is demonstrated via extensive experiments on both simulated and real data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.04201v3-abstract-full').style.display = 'none'; document.getElementById('2108.04201v3-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> 25 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2021. </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 American Statistical Association, 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/2106.05925">arXiv:2106.05925</a> <span> [<a href="https://arxiv.org/pdf/2106.05925">pdf</a>, <a href="https://arxiv.org/format/2106.05925">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"> Online Debiased Lasso for Streaming Data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Ruijian Han</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Lan Luo</a>, <a href="/search/math?searchtype=author&query=Lin%2C+Y">Yuanyuan Lin</a>, <a href="/search/math?searchtype=author&query=Huang%2C+J">Jian Huang</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="2106.05925v2-abstract-short" style="display: inline;"> We propose an online debiased lasso (ODL) method for statistical inference in high-dimensional linear models with streaming data. The proposed ODL consists of an efficient computational algorithm for streaming data and approximately normal estimators for the regression coefficients. Its implementation only requires the availability of the current data batch in the data stream and sufficient statis… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.05925v2-abstract-full').style.display = 'inline'; document.getElementById('2106.05925v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.05925v2-abstract-full" style="display: none;"> We propose an online debiased lasso (ODL) method for statistical inference in high-dimensional linear models with streaming data. The proposed ODL consists of an efficient computational algorithm for streaming data and approximately normal estimators for the regression coefficients. Its implementation only requires the availability of the current data batch in the data stream and sufficient statistics of the historical data at each stage of the analysis. A dynamic procedure is developed to select and update the tuning parameters upon the arrival of each new data batch so that we can adjust the amount of regularization adaptively along the data stream. The asymptotic normality of the ODL estimator is established under the conditions similar to those in an offline setting and mild conditions on the size of data batches in the stream, which provides theoretical justification for the proposed online statistical inference procedure. We conduct extensive numerical experiments to evaluate the performance of ODL. These experiments demonstrate the effectiveness of our algorithm and support the theoretical results. An air quality dataset and an index fund dataset from Hong Kong Stock Exchange are analyzed to illustrate the application of the proposed method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.05925v2-abstract-full').style.display = 'none'; document.getElementById('2106.05925v2-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> 18 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </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">Ruijian Han and Lan Luo contributed equally to this work. Co-corresponding authors: Yuanyuan Lin (Email: ylin@sta.cuhk.edu.hk) and Jian Huang (Email: jian-huang@uiowa.edu)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.11686">arXiv:2012.11686</a> <span> [<a href="https://arxiv.org/pdf/2012.11686">pdf</a>, <a href="https://arxiv.org/ps/2012.11686">ps</a>, <a href="https://arxiv.org/format/2012.11686">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="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> A Polynomial Roth Theorem for Corners in Finite Fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Lacey%2C+M+T">Michael T Lacey</a>, <a href="/search/math?searchtype=author&query=Yang%2C+F">Fan Yang</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="2012.11686v2-abstract-short" style="display: inline;"> We prove a Roth type theorem for polynomial corners in the finite field setting. Let $蠁_1$ and $蠁_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $ A \subset \mathbb F_p \times \mathbb F_p$ with $ \lvert A\rvert > p ^{2 - \frac1{16}} $ contains three points $ (x_1, x_2) , (x_1 + 蠁_1 (y), x_2), (x_1, x_2 + 蠁_2 (y))$. The study of these questions on… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.11686v2-abstract-full').style.display = 'inline'; document.getElementById('2012.11686v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.11686v2-abstract-full" style="display: none;"> We prove a Roth type theorem for polynomial corners in the finite field setting. Let $蠁_1$ and $蠁_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $ A \subset \mathbb F_p \times \mathbb F_p$ with $ \lvert A\rvert > p ^{2 - \frac1{16}} $ contains three points $ (x_1, x_2) , (x_1 + 蠁_1 (y), x_2), (x_1, x_2 + 蠁_2 (y))$. The study of these questions on $ \mathbb F_p$ was started by Bourgain and Chang. Our Theorem adapts the argument of Dong, Li and Sawin, in particular relying upon deep Weil type inequalities established by N. Katz. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.11686v2-abstract-full').style.display = 'none'; document.getElementById('2012.11686v2-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 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2020. </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. Minor changes for the final version of the paper</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.09996">arXiv:2012.09996</a> <span> [<a href="https://arxiv.org/pdf/2012.09996">pdf</a>, <a href="https://arxiv.org/format/2012.09996">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="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rungang Han</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yuetian Luo</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Miaoyan Wang</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="2012.09996v4-abstract-short" style="display: inline;"> High-order clustering aims to identify heterogeneous substructures in multiway datasets that arise commonly in neuroimaging, genomics, social network studies, etc. The non-convex and discontinuous nature of this problem pose significant challenges in both statistics and computation. In this paper, we propose a tensor block model and the computationally efficient methods, \emph{high-order Lloyd alg… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.09996v4-abstract-full').style.display = 'inline'; document.getElementById('2012.09996v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.09996v4-abstract-full" style="display: none;"> High-order clustering aims to identify heterogeneous substructures in multiway datasets that arise commonly in neuroimaging, genomics, social network studies, etc. The non-convex and discontinuous nature of this problem pose significant challenges in both statistics and computation. In this paper, we propose a tensor block model and the computationally efficient methods, \emph{high-order Lloyd algorithm} (HLloyd), and high-order spectral clustering (HSC), for high-order clustering. The convergence guarantees and statistical optimality are established for the proposed procedure under a mild sub-Gaussian noise assumption. Under the Gaussian tensor block model, we completely characterize the statistical-computational trade-off for achieving high-order exact clustering based on three different signal-to-noise ratio regimes. The analysis relies on new techniques of high-order spectral perturbation analysis and a ``singular-value-gap-free'' error bound in tensor estimation, which are substantially different from the matrix spectral analyses in the literature. Finally, we show the merits of the proposed procedures via extensive experiments on both synthetic and real datasets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.09996v4-abstract-full').style.display = 'none'; document.getElementById('2012.09996v4-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 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2020. </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/2008.12434">arXiv:2008.12434</a> <span> [<a href="https://arxiv.org/pdf/2008.12434">pdf</a>, <a href="https://arxiv.org/ps/2008.12434">ps</a>, <a href="https://arxiv.org/format/2008.12434">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="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> On the Non-Asymptotic Concentration of Heteroskedastic Wishart-type Matrix </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cai%2C+T+T">T. Tony Cai</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Rungang Han</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="2008.12434v2-abstract-short" style="display: inline;"> This paper focuses on the non-asymptotic concentration of the heteroskedastic Wishart-type matrices. Suppose $Z$ is a $p_1$-by-$p_2$ random matrix and $Z_{ij} \sim N(0,蟽_{ij}^2)$ independently, we prove the expected spectral norm of Wishart matrix deviations (i.e., $\mathbb{E} \left\|ZZ^\top - \mathbb{E} ZZ^\top\right\|$) is upper bounded by \begin{equation*} \begin{split} (1+蔚)\left\{2蟽_C蟽_R… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.12434v2-abstract-full').style.display = 'inline'; document.getElementById('2008.12434v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.12434v2-abstract-full" style="display: none;"> This paper focuses on the non-asymptotic concentration of the heteroskedastic Wishart-type matrices. Suppose $Z$ is a $p_1$-by-$p_2$ random matrix and $Z_{ij} \sim N(0,蟽_{ij}^2)$ independently, we prove the expected spectral norm of Wishart matrix deviations (i.e., $\mathbb{E} \left\|ZZ^\top - \mathbb{E} ZZ^\top\right\|$) is upper bounded by \begin{equation*} \begin{split} (1+蔚)\left\{2蟽_C蟽_R + 蟽_C^2 + C蟽_R蟽_*\sqrt{\log(p_1 \wedge p_2)} + C蟽_*^2\log(p_1 \wedge p_2)\right\}, \end{split} \end{equation*} where $蟽_C^2 := \max_j \sum_{i=1}^{p_1}蟽_{ij}^2$, $蟽_R^2 := \max_i \sum_{j=1}^{p_2}蟽_{ij}^2$ and $蟽_*^2 := \max_{i,j}蟽_{ij}^2$. A minimax lower bound is developed that matches this upper bound. Then, we derive the concentration inequalities, moments, and tail bounds for the heteroskedastic Wishart-type matrix under more general distributions, such as sub-Gaussian and heavy-tailed distributions. Next, we consider the cases where $Z$ has homoskedastic columns or rows (i.e., $蟽_{ij} \approx 蟽_i$ or $蟽_{ij} \approx 蟽_j$) and derive the rate-optimal Wishart-type concentration bounds. Finally, we apply the developed tools to identify the sharp signal-to-noise ratio threshold for consistent clustering in the heteroskedastic clustering problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.12434v2-abstract-full').style.display = 'none'; document.getElementById('2008.12434v2-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, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </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">Electronic Journal of Probability, 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/2008.06329">arXiv:2008.06329</a> <span> [<a href="https://arxiv.org/pdf/2008.06329">pdf</a>, <a href="https://arxiv.org/format/2008.06329">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> </div> <p class="title is-5 mathjax"> Honeycomb structures in magnetic fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Becker%2C+S">Simon Becker</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Jitomirskaya%2C+S">Svetlana Jitomirskaya</a>, <a href="/search/math?searchtype=author&query=Zworski%2C+M">Maciej Zworski</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="2008.06329v1-abstract-short" style="display: inline;"> We consider reduced-dimensionality models of honeycomb lattices in magnetic fields and report results about the spectrum, the density of states, self-similarity, and metal/insulator transitions under disorder. We perform a spectral analysis by which we discover a fractal Cantor spectrum for irrational magnetic flux through a honeycomb, prove the existence of zero energy Dirac cones for each ration… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.06329v1-abstract-full').style.display = 'inline'; document.getElementById('2008.06329v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.06329v1-abstract-full" style="display: none;"> We consider reduced-dimensionality models of honeycomb lattices in magnetic fields and report results about the spectrum, the density of states, self-similarity, and metal/insulator transitions under disorder. We perform a spectral analysis by which we discover a fractal Cantor spectrum for irrational magnetic flux through a honeycomb, prove the existence of zero energy Dirac cones for each rational flux, obtain an explicit expansion of the density of states near the conical points, and show the existence of mobility edges under Anderson-type disorder. Our results give a precise description of de Haas-van Alphen and Quantum Hall effects, and provide quantitative estimates on transport properties. In particular, our findings explain experimentally observed asymmetry phenomena by going beyond the perfect cone approximation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.06329v1-abstract-full').style.display = 'none'; document.getElementById('2008.06329v1-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, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.01312">arXiv:2008.01312</a> <span> [<a href="https://arxiv.org/pdf/2008.01312">pdf</a>, <a href="https://arxiv.org/format/2008.01312">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"> A Schatten-$q$ Low-rank Matrix Perturbation Analysis via Perturbation Projection Error Bound </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=Han%2C+R">Rungang Han</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="2008.01312v3-abstract-short" style="display: inline;"> This paper studies the Schatten-$q$ error of low-rank matrix estimation by singular value decomposition under perturbation. We specifically establish a perturbation bound on the low-rank matrix estimation via a perturbation projection error bound. Then, we establish lower bounds to justify the tightness of the upper bound on the low-rank matrix estimation error. We further develop a user-friendly… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.01312v3-abstract-full').style.display = 'inline'; document.getElementById('2008.01312v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.01312v3-abstract-full" style="display: none;"> This paper studies the Schatten-$q$ error of low-rank matrix estimation by singular value decomposition under perturbation. We specifically establish a perturbation bound on the low-rank matrix estimation via a perturbation projection error bound. Then, we establish lower bounds to justify the tightness of the upper bound on the low-rank matrix estimation error. We further develop a user-friendly sin$螛$ bound for singular subspace perturbation based on the matrix perturbation projection error bound. Finally, we demonstrate the advantage of our results over the ones in the literature by simulation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.01312v3-abstract-full').style.display = 'none'; document.getElementById('2008.01312v3-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 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </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">Accepted to Linear Algebra and its Applications</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2004.06189">arXiv:2004.06189</a> <span> [<a href="https://arxiv.org/pdf/2004.06189">pdf</a>, <a href="https://arxiv.org/format/2004.06189">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</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"> Density of states and Delocalization for discrete magnetic random Schr枚dinger operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Becker%2C+S">Simon Becker</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</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="2004.06189v2-abstract-short" style="display: inline;"> We study discrete magnetic random Schr枚dinger operators on the square and honeycomb lattice. For the non-random magnetic operator on the hexagonal lattice with any rational magnetic flux, we show that the middle two dispersion surfaces exhibit Dirac cones. We then derive an asymptotic expansion for the density of states on the honeycomb lattice for oscillations of arbitrary rational magnetic flux.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.06189v2-abstract-full').style.display = 'inline'; document.getElementById('2004.06189v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2004.06189v2-abstract-full" style="display: none;"> We study discrete magnetic random Schr枚dinger operators on the square and honeycomb lattice. For the non-random magnetic operator on the hexagonal lattice with any rational magnetic flux, we show that the middle two dispersion surfaces exhibit Dirac cones. We then derive an asymptotic expansion for the density of states on the honeycomb lattice for oscillations of arbitrary rational magnetic flux. This allows us, as a corollary, to rigorously study the quantum Hall effect and conclude dynamical delocalization close to the conical point under disorder. We obtain similar results for the discrete random Schr枚dinger operator on the $\mathbb Z^2$-lattice with weak magnetic fields, close to the bottom and top of its spectrum. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.06189v2-abstract-full').style.display = 'none'; document.getElementById('2004.06189v2-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 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2020. </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">comments 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/2002.11255">arXiv:2002.11255</a> <span> [<a href="https://arxiv.org/pdf/2002.11255">pdf</a>, <a href="https://arxiv.org/format/2002.11255">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="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"> An Optimal Statistical and Computational Framework for Generalized Tensor Estimation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rungang Han</a>, <a href="/search/math?searchtype=author&query=Willett%2C+R">Rebecca Willett</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="2002.11255v2-abstract-short" style="display: inline;"> This paper describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator consists of finding a low-rank tensor fit to the data under generalized parametric models. To overcome the difficulty of non-convexity in these problems, we int… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.11255v2-abstract-full').style.display = 'inline'; document.getElementById('2002.11255v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.11255v2-abstract-full" style="display: none;"> This paper describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator consists of finding a low-rank tensor fit to the data under generalized parametric models. To overcome the difficulty of non-convexity in these problems, we introduce a unified approach of projected gradient descent that adapts to the underlying low-rank structure. Under mild conditions on the loss function, we establish both an upper bound on statistical error and the linear rate of computational convergence through a general deterministic analysis. Then we further consider a suite of generalized tensor estimation problems, including sub-Gaussian tensor PCA, tensor regression, and Poisson and binomial tensor PCA. We prove that the proposed algorithm achieves the minimax optimal rate of convergence in estimation error. Finally, we demonstrate the superiority of the proposed framework via extensive experiments on both simulated and real data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.11255v2-abstract-full').style.display = 'none'; document.getElementById('2002.11255v2-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 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.08853">arXiv:2002.08853</a> <span> [<a href="https://arxiv.org/pdf/2002.08853">pdf</a>, <a href="https://arxiv.org/format/2002.08853">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="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> A General Pairwise Comparison Model for Extremely Sparse Networks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Ruijian Han</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Y">Yiming Xu</a>, <a href="/search/math?searchtype=author&query=Chen%2C+K">Kani 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="2002.08853v3-abstract-short" style="display: inline;"> Statistical inference using pairwise comparison data is an effective approach to analyzing large-scale sparse networks. In this paper, we propose a general framework to model the mutual interactions in a network, which enjoys ample flexibility in terms of model parametrization. Under this setup, we show that the maximum likelihood estimator for the latent score vector of the subjects is uniformly… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.08853v3-abstract-full').style.display = 'inline'; document.getElementById('2002.08853v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.08853v3-abstract-full" style="display: none;"> Statistical inference using pairwise comparison data is an effective approach to analyzing large-scale sparse networks. In this paper, we propose a general framework to model the mutual interactions in a network, which enjoys ample flexibility in terms of model parametrization. Under this setup, we show that the maximum likelihood estimator for the latent score vector of the subjects is uniformly consistent under a near-minimal condition on network sparsity. This condition is sharp in terms of the leading order asymptotics describing the sparsity. Our analysis utilizes a novel chaining technique and illustrates an important connection between graph topology and model consistency. Our results guarantee that the maximum likelihood estimator is justified for estimation in large-scale pairwise comparison networks where data are asymptotically deficient. Simulation studies are provided in support of our theoretical findings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.08853v3-abstract-full').style.display = 'none'; document.getElementById('2002.08853v3-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 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </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">33 pages, 1 figure, 1 table</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1910.14630">arXiv:1910.14630</a> <span> [<a href="https://arxiv.org/pdf/1910.14630">pdf</a>, <a href="https://arxiv.org/ps/1910.14630">ps</a>, <a href="https://arxiv.org/format/1910.14630">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> </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/10.1007/s00041-020-09748-4">10.1007/s00041-020-09748-4 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Improving estimates for discrete polynomial averages </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Kova%C4%8D%2C+V">Vjekoslav Kova膷</a>, <a href="/search/math?searchtype=author&query=Lacey%2C+M">Michael Lacey</a>, <a href="/search/math?searchtype=author&query=Madrid%2C+J">Jos茅 Madrid</a>, <a href="/search/math?searchtype=author&query=Yang%2C+F">Fan Yang</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="1910.14630v3-abstract-short" style="display: inline;"> For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving inequality \begin{equation*} \|A_N f\|_{\ell^q(\mathbb{Z})} \lesssim_{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})} \|f\|_{\ell^p(\mathbb{Z})}, \qquad N \in\mathbb{N}, \end… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.14630v3-abstract-full').style.display = 'inline'; document.getElementById('1910.14630v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1910.14630v3-abstract-full" style="display: none;"> For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving inequality \begin{equation*} \|A_N f\|_{\ell^q(\mathbb{Z})} \lesssim_{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})} \|f\|_{\ell^p(\mathbb{Z})}, \qquad N \in\mathbb{N}, \end{equation*} where $1\leq p \leq q \leq \infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.14630v3-abstract-full').style.display = 'none'; document.getElementById('1910.14630v3-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 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2019. </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">10 pages. This version combines arXiv:1910.12448 by J. Madrid with arXiv:1910.14630v1 by the remaining four authors. To appear in JFAA</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Fourier Anal. Appl. 26 (2020), Article 42 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.03995">arXiv:1909.03995</a> <span> [<a href="https://arxiv.org/pdf/1909.03995">pdf</a>, <a href="https://arxiv.org/ps/1909.03995">ps</a>, <a href="https://arxiv.org/format/1909.03995">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</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/10.1093/imrn/rnw279">10.1093/imrn/rnw279 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Absence of point spectrum for the self-dual extended Harper's model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</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="1909.03995v1-abstract-short" style="display: inline;"> We give a simple proof of absence of point spectrum for the self-dual extended Harper's model. We get a sharp result which improves that of Avila-Jitomirskaya-Marx in the isotropic self-dual regime. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.03995v1-abstract-full" style="display: none;"> We give a simple proof of absence of point spectrum for the self-dual extended Harper's model. We get a sharp result which improves that of Avila-Jitomirskaya-Marx in the isotropic self-dual regime. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.03995v1-abstract-full').style.display = 'none'; document.getElementById('1909.03995v1-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 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 47B36 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> IMRN (2017) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.02883">arXiv:1909.02883</a> <span> [<a href="https://arxiv.org/pdf/1909.02883">pdf</a>, <a href="https://arxiv.org/ps/1909.02883">ps</a>, <a href="https://arxiv.org/format/1909.02883">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> </div> </div> <p class="title is-5 mathjax"> Averages Along the Primes: Improving and Sparse Bounds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Krause%2C+B">Ben Krause</a>, <a href="/search/math?searchtype=author&query=Lacey%2C+M">Michael Lacey</a>, <a href="/search/math?searchtype=author&query=Yang%2C+F">Fan Yang</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="1909.02883v2-abstract-short" style="display: inline;"> Consider averages along the prime integers $ \mathbb P $ given by \begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*} These averages satisfy a uniform scale-free $ \ell ^{p}$-improving estimate. For all $ 1< p < 2$, there is a constant $ C_p$ so that for all integer $ N$ and functions $ f$ supported on $ [0,N]$, there holds \begin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.02883v2-abstract-full').style.display = 'inline'; document.getElementById('1909.02883v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.02883v2-abstract-full" style="display: none;"> Consider averages along the prime integers $ \mathbb P $ given by \begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*} These averages satisfy a uniform scale-free $ \ell ^{p}$-improving estimate. For all $ 1< p < 2$, there is a constant $ C_p$ so that for all integer $ N$ and functions $ f$ supported on $ [0,N]$, there holds \begin{equation*} N ^{-1/p' }\lVert \mathcal{A}_N f\rVert_{\ell^{p'}} \leq C_p N ^{- 1/p} \lVert f\rVert_{\ell^p}. \end{equation*} The maximal function $ \mathcal{A}^{\ast} f =\sup_{N} \lvert \mathcal{A}_N f \rvert$ satisfies $ (p,p)$ sparse bounds for all $ 1< p < 2$. The latter are the natural variants of the scale-free bounds. As a corollary, $ \mathcal{A}^{\ast} $ is bounded on $ \ell ^{p} (w)$, for all weights $ w$ in the Muckenhoupt $A_p$ class. No prior weighted inequalities for $ \mathcal{A}^{\ast} $ were known. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.02883v2-abstract-full').style.display = 'none'; document.getElementById('1909.02883v2-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> 19 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2019. </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">13 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/1907.05734">arXiv:1907.05734</a> <span> [<a href="https://arxiv.org/pdf/1907.05734">pdf</a>, <a href="https://arxiv.org/ps/1907.05734">ps</a>, <a href="https://arxiv.org/format/1907.05734">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> </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/10.2140/tunis.2021.3.517">10.2140/tunis.2021.3.517 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Averages along the Square Integers: $\ell^p$ improving and Sparse Inequalities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Lacey%2C+M+T">Michael T Lacey</a>, <a href="/search/math?searchtype=author&query=Yang%2C+F">Fan Yang</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="1907.05734v2-abstract-short" style="display: inline;"> Let $f\in \ell^2(\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\frac{1}{N}\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \ell ^{p}$-improving estimate, for $ 3/2 < p \leq 2$: \begin{equation*} N ^{-2/p'} \lVert A_N f \rVert _{ p'} \lesssim N ^{-2/p} \lVert f\rVert _{\ell ^{p}}, \end{equation*} provided $ f$ is supported in some inter… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1907.05734v2-abstract-full').style.display = 'inline'; document.getElementById('1907.05734v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1907.05734v2-abstract-full" style="display: none;"> Let $f\in \ell^2(\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\frac{1}{N}\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \ell ^{p}$-improving estimate, for $ 3/2 < p \leq 2$: \begin{equation*} N ^{-2/p'} \lVert A_N f \rVert _{ p'} \lesssim N ^{-2/p} \lVert f\rVert _{\ell ^{p}}, \end{equation*} provided $ f$ is supported in some interval of length $ N ^2 $, and $ p' =\frac{p} {p-1}$ is the conjugate index. The inequality above fails for $ 1< p < 3/2$. The maximal function $ A f = \sup _{N\geq 1} |A_Nf| $ satisfies a similar sparse bound. Novel weighted and vector valued inequalities for $ A$ follow. A critical step in the proof requires the control of a logarithmic average over $ q$ of a function $G(q,x)$ counting the number of square roots of $x$ mod $q$. One requires an estimate uniform in $x$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1907.05734v2-abstract-full').style.display = 'none'; document.getElementById('1907.05734v2-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, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 July, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2019. </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. To appear in Tunisian Journal of Mathematics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Tunisian J. Math. 3 (2021) 517-550 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.03794">arXiv:1811.03794</a> <span> [<a href="https://arxiv.org/pdf/1811.03794">pdf</a>, <a href="https://arxiv.org/ps/1811.03794">ps</a>, <a href="https://arxiv.org/format/1811.03794">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Physics and Society">physics.soc-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</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.cnsns.2019.01.012">10.1016/j.cnsns.2019.01.012 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Tetradic motif profiles of horizontal visibility graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Wen-Jie Xie</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Rui-Qi Han</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+W">Wei-Xing 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="1811.03794v1-abstract-short" style="display: inline;"> Network motif analysis is a useful tool for the investigation of complex networks. We study the profiles of tetradic motifs in horizontal visibility graphs (HVGs) converted from multifractal binomial measures, fractional Gaussian noises, and heartbeat rates. The profiles of tetradic motifs contains the spatial information (visibility) and temporal information (relative magnitude) among the data po… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.03794v1-abstract-full').style.display = 'inline'; document.getElementById('1811.03794v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.03794v1-abstract-full" style="display: none;"> Network motif analysis is a useful tool for the investigation of complex networks. We study the profiles of tetradic motifs in horizontal visibility graphs (HVGs) converted from multifractal binomial measures, fractional Gaussian noises, and heartbeat rates. The profiles of tetradic motifs contains the spatial information (visibility) and temporal information (relative magnitude) among the data points in the corresponding time series. For multifractal binomial measures, the occurrence frequencies of the tetradic motifs are determined, which converge to a constant vector $(2/3,0,8/99,8/33,1/99,0)$. For fractional Gaussian noises, the motif occurrence frequencies are found to depend nonlinearly on the Hurst exponent and the length of time series. These findings suggest the potential ability of tetradic motif profiles in distinguishing different types of time series. Finally, we apply the tetradic motif analysis to heartbeat rates of healthy subjects, congestive heart failure (CHF) subjects, and atrial fibrillation (AF) subjects. Different subjects can be distinguished from the occurrence frequencies of tetradic motifs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.03794v1-abstract-full').style.display = 'none'; document.getElementById('1811.03794v1-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 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </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">9 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Commun Nonlinear Sci Numer Simulat 72 (2019) 544-551 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.01923">arXiv:1811.01923</a> <span> [<a href="https://arxiv.org/pdf/1811.01923">pdf</a>, <a href="https://arxiv.org/ps/1811.01923">ps</a>, <a href="https://arxiv.org/format/1811.01923">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> </div> </div> <p class="title is-5 mathjax"> Weighted Estimates for One Sided Martingale Transforms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+W">Wei Chen</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Lacey%2C+M+T">Michael T. Lacey</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="1811.01923v1-abstract-short" style="display: inline;"> Let $ Tf =\sum_{ I} \varepsilon_I \langle f,h_{I^+}\rangle h_{I^-}$. Here, $ \lvert \varepsilon _I\rvert=1 $, and $ h_J$ is the Haar function defined on dyadic interval $ J$. We show that, for instance, \begin{equation*} \lVert T \rVert _{L ^{2} (w) \to L ^{2} (w)} \lesssim [w] _{A_2 ^{+}} . \end{equation*} Above, we use the one sided $ A_2$ characteristic for the weight $ w$. This is an instance… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.01923v1-abstract-full').style.display = 'inline'; document.getElementById('1811.01923v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.01923v1-abstract-full" style="display: none;"> Let $ Tf =\sum_{ I} \varepsilon_I \langle f,h_{I^+}\rangle h_{I^-}$. Here, $ \lvert \varepsilon _I\rvert=1 $, and $ h_J$ is the Haar function defined on dyadic interval $ J$. We show that, for instance, \begin{equation*} \lVert T \rVert _{L ^{2} (w) \to L ^{2} (w)} \lesssim [w] _{A_2 ^{+}} . \end{equation*} Above, we use the one sided $ A_2$ characteristic for the weight $ w$. This is an instance of a one sided $A_2$ conjecture. Our proof of this fact is difficult, as the very quick known proofs of the $A_2$ theorem do not seem to apply in the one sided setting. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.01923v1-abstract-full').style.display = 'none'; document.getElementById('1811.01923v1-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 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </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> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1809.01796">arXiv:1809.01796</a> <span> [<a href="https://arxiv.org/pdf/1809.01796">pdf</a>, <a href="https://arxiv.org/format/1809.01796">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> <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"> Optimal Sparse Singular Value Decomposition for High-dimensional High-order Data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+A">Anru Zhang</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Rungang Han</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="1809.01796v2-abstract-short" style="display: inline;"> In this article, we consider the sparse tensor singular value decomposition, which aims for dimension reduction on high-dimensional high-order data with certain sparsity structure. A method named Sparse Tensor Alternating Thresholding for Singular Value Decomposition (STAT-SVD) is proposed. The proposed procedure features a novel double projection \& thresholding scheme, which provides a sharp cri… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1809.01796v2-abstract-full').style.display = 'inline'; document.getElementById('1809.01796v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1809.01796v2-abstract-full" style="display: none;"> In this article, we consider the sparse tensor singular value decomposition, which aims for dimension reduction on high-dimensional high-order data with certain sparsity structure. A method named Sparse Tensor Alternating Thresholding for Singular Value Decomposition (STAT-SVD) is proposed. The proposed procedure features a novel double projection \& thresholding scheme, which provides a sharp criterion for thresholding in each iteration. Compared with regular tensor SVD model, STAT-SVD permits more robust estimation under weaker assumptions. Both the upper and lower bounds for estimation accuracy are developed. The proposed procedure is shown to be minimax rate-optimal in a general class of situations. Simulation studies show that STAT-SVD performs well under a variety of configurations. We also illustrate the merits of the proposed procedure on a longitudinal tensor dataset on European country mortality rates. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1809.01796v2-abstract-full').style.display = 'none'; document.getElementById('1809.01796v2-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> 7 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 September, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2018. </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">73 pages; typo fixed</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1807.00233">arXiv:1807.00233</a> <span> [<a href="https://arxiv.org/pdf/1807.00233">pdf</a>, <a href="https://arxiv.org/ps/1807.00233">ps</a>, <a href="https://arxiv.org/format/1807.00233">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</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"> Weyl sums and the Lyapunov exponent for the skew-shift Schr枚dinger cocycle </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Lemm%2C+M">Marius Lemm</a>, <a href="/search/math?searchtype=author&query=Schlag%2C+W">Wilhelm Schlag</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="1807.00233v1-abstract-short" style="display: inline;"> We study the one-dimensional discrete Schr枚dinger operator with the skew-shift potential $2位\cos\left(2蟺\left(\binom{j}{2} 蠅+jy+x\right)\right)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $位>0$. In this paper, we introduce a novel perturbative approach for studying the zero-energy L… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1807.00233v1-abstract-full').style.display = 'inline'; document.getElementById('1807.00233v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1807.00233v1-abstract-full" style="display: none;"> We study the one-dimensional discrete Schr枚dinger operator with the skew-shift potential $2位\cos\left(2蟺\left(\binom{j}{2} 蠅+jy+x\right)\right)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $位>0$. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent $L(位)$ at small $位$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $L(位)$ is fully consistent with $L(位)$ being positive and satisfying the usual Figotin-Pastur type asymptotics $L(位)\sim C位^2$ as $位\to 0$. The analogous quantity behaves completely differently in the Almost-Mathieu model, whose zero-energy Lyapunov exponent vanishes for $位<1$. The main technical work consists in establishing good lower bounds on the exponential sums (quadratic Weyl sums) that appear in our perturbation series. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1807.00233v1-abstract-full').style.display = 'none'; document.getElementById('1807.00233v1-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> 30 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2018. </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">33 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/1806.01988">arXiv:1806.01988</a> <span> [<a href="https://arxiv.org/pdf/1806.01988">pdf</a>, <a href="https://arxiv.org/ps/1806.01988">ps</a>, <a href="https://arxiv.org/format/1806.01988">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Discrete Bethe--Sommerfeld Conjecture for Triangular, Square, and Hexagonal Lattices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fillman%2C+J">Jake Fillman</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</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="1806.01988v1-abstract-short" style="display: inline;"> We study discrete Schr枚dinger operators on the graphs corresponding to the triangular lattice, the hexagonal lattice, and the square lattice with next-nearest neighbor interactions. For each of these lattice geometries, we analyze the behavior of small periodic potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we describe sharp arithmetic criter… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.01988v1-abstract-full').style.display = 'inline'; document.getElementById('1806.01988v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1806.01988v1-abstract-full" style="display: none;"> We study discrete Schr枚dinger operators on the graphs corresponding to the triangular lattice, the hexagonal lattice, and the square lattice with next-nearest neighbor interactions. For each of these lattice geometries, we analyze the behavior of small periodic potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we describe sharp arithmetic criteria on the periods that ensure that no gaps open, and we characterize those energies at which gaps may open in the perturbative regime. In all three cases, we provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.01988v1-abstract-full').style.display = 'none'; document.getElementById('1806.01988v1-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 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2018. </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">39 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/1805.04994">arXiv:1805.04994</a> <span> [<a href="https://arxiv.org/pdf/1805.04994">pdf</a>, <a href="https://arxiv.org/format/1805.04994">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="Analysis of PDEs">math.AP</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/10.2140/apde.2020.13.813">10.2140/apde.2020.13.813 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A higher dimensional Bourgain-Dyatlov fractal uncertainty principle </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Schlag%2C+W">Wilhelm Schlag</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="1805.04994v2-abstract-short" style="display: inline;"> We establish a version of the fractal uncertainty principle, obtained by Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many products of $未$-regular sets in one dimension, but relative to arbitrary axes. Our results remain true if $Y$ is distorted by diffeomorphisms. Our method combines the origina… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.04994v2-abstract-full').style.display = 'inline'; document.getElementById('1805.04994v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1805.04994v2-abstract-full" style="display: none;"> We establish a version of the fractal uncertainty principle, obtained by Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many products of $未$-regular sets in one dimension, but relative to arbitrary axes. Our results remain true if $Y$ is distorted by diffeomorphisms. Our method combines the original approach by Bourgain and Dyatlov, in the more quantitative 2017 rendition by Jin and Zhang, with Cartan set techniques. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.04994v2-abstract-full').style.display = 'none'; document.getElementById('1805.04994v2-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 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2018. </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">44 pages, arguments simplified and 2 figures added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Analysis & PDE 13 (2020) 813-863 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1804.04322">arXiv:1804.04322</a> <span> [<a href="https://arxiv.org/pdf/1804.04322">pdf</a>, <a href="https://arxiv.org/ps/1804.04322">ps</a>, <a href="https://arxiv.org/format/1804.04322">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> <p class="title is-5 mathjax"> Spectral Dimension for $尾$-almost periodic singular Jacobi operators and the extended Harper's model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Yang%2C+F">Fan Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+S">Shiwen 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="1804.04322v2-abstract-short" style="display: inline;"> We study fractal dimension properties of singular Jacobi operators. We prove quantitative lower spectral/quantum dynamical bounds for general operators with strong repetition properties and controlled singularities. For analytic quasiperiodic Jacobi operators in the positive Lyapunov exponent regime, we obtain a sharp arithmetic criterion of full spectral dimensionality. The applications include t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.04322v2-abstract-full').style.display = 'inline'; document.getElementById('1804.04322v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1804.04322v2-abstract-full" style="display: none;"> We study fractal dimension properties of singular Jacobi operators. We prove quantitative lower spectral/quantum dynamical bounds for general operators with strong repetition properties and controlled singularities. For analytic quasiperiodic Jacobi operators in the positive Lyapunov exponent regime, we obtain a sharp arithmetic criterion of full spectral dimensionality. The applications include the extended Harper's model where we obtain arithmetic results on spectral dimensions and quantum dynamical exponents. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.04322v2-abstract-full').style.display = 'none'; document.getElementById('1804.04322v2-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 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1803.02035">arXiv:1803.02035</a> <span> [<a href="https://arxiv.org/pdf/1803.02035">pdf</a>, <a href="https://arxiv.org/ps/1803.02035">ps</a>, <a href="https://arxiv.org/format/1803.02035">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</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"> Large deviation estimates and H枚lder regularity of the Lyapunov exponents for quasi-periodic Schr枚dinger cocycles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+S">Shiwen 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="1803.02035v2-abstract-short" style="display: inline;"> We consider one-dimensional quasi-periodic Schr枚dinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates which lead to optimal H枚lder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.02035v2-abstract-full').style.display = 'inline'; document.getElementById('1803.02035v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1803.02035v2-abstract-full" style="display: none;"> We consider one-dimensional quasi-periodic Schr枚dinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates which lead to optimal H枚lder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.02035v2-abstract-full').style.display = 'none'; document.getElementById('1803.02035v2-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 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2018. </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">International Mathematics Research Notices</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1803.02034">arXiv:1803.02034</a> <span> [<a href="https://arxiv.org/pdf/1803.02034">pdf</a>, <a href="https://arxiv.org/ps/1803.02034">ps</a>, <a href="https://arxiv.org/format/1803.02034">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link 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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Effective multi-scale approach to the Schr枚dinger cocycle over a skew shift base </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Lemm%2C+M">Marius Lemm</a>, <a href="/search/math?searchtype=author&query=Schlag%2C+W">Wilhelm Schlag</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="1803.02034v1-abstract-short" style="display: inline;"> We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schr枚dinger cocycle over a skew shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman's subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large deviation estimates at tha… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.02034v1-abstract-full').style.display = 'inline'; document.getElementById('1803.02034v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1803.02034v1-abstract-full" style="display: none;"> We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schr枚dinger cocycle over a skew shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman's subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schr枚dinger operators with deterministic potentials, based on large deviation estimates and the avalanche principle, effective. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.02034v1-abstract-full').style.display = 'none'; document.getElementById('1803.02034v1-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 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1803.00988">arXiv:1803.00988</a> <span> [<a href="https://arxiv.org/pdf/1803.00988">pdf</a>, <a href="https://arxiv.org/format/1803.00988">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</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/10.1007/s00222-019-00916-y">10.1007/s00222-019-00916-y <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Cantor spectrum of graphene in magnetic fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Becker%2C+S">Simon Becker</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Jitomirskaya%2C+S">Svetlana Jitomirskaya</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="1803.00988v1-abstract-short" style="display: inline;"> We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux $桅/2蟺$ through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1803.00988v1-abstract-full" style="display: none;"> We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux $桅/2蟺$ through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.00988v1-abstract-full').style.display = 'none'; document.getElementById('1803.00988v1-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 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2018. </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">55 pages, 8 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/1801.04621">arXiv:1801.04621</a> <span> [<a href="https://arxiv.org/pdf/1801.04621">pdf</a>, <a href="https://arxiv.org/format/1801.04621">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> </div> </div> <p class="title is-5 mathjax"> Computing Shape DNA using the closest point method </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rachel Han</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="1801.04621v1-abstract-short" style="display: inline;"> We demonstrate an application of the closest point method where the truncated spectrum of the Laplace--Beltrami operator of an object is used to identify the object. The effectiveness of the method is analyzed as well as the default algorithm, `eigs', in MATLAB which computes the eigenvalues of a given matrix. We also cluster "similar" objects via multi-dimensional scaling algorithm and empiricall… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.04621v1-abstract-full').style.display = 'inline'; document.getElementById('1801.04621v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1801.04621v1-abstract-full" style="display: none;"> We demonstrate an application of the closest point method where the truncated spectrum of the Laplace--Beltrami operator of an object is used to identify the object. The effectiveness of the method is analyzed as well as the default algorithm, `eigs', in MATLAB which computes the eigenvalues of a given matrix. We also cluster "similar" objects via multi-dimensional scaling algorithm and empirically measure its effectiveness. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.04621v1-abstract-full').style.display = 'none'; document.getElementById('1801.04621v1-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, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1712.01782">arXiv:1712.01782</a> <span> [<a href="https://arxiv.org/pdf/1712.01782">pdf</a>, <a href="https://arxiv.org/ps/1712.01782">ps</a>, <a href="https://arxiv.org/format/1712.01782">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</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"> Generic continuous spectrum for multi-dimensional quasi periodic Schr枚dinger operators with rough potentials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Yang%2C+F">Fan Yang</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="1712.01782v1-abstract-short" style="display: inline;"> We study the multi-dimensional operator $(H_x u)_n=\sum_{|m-n|=1}u_{m}+f(T^n(x))u_n$, where $T$ is the shift of the torus $\T^d$. When $d=2$, we show the spectrum of $H_x$ is almost surely purely continuous for a.e. $伪$ and generic continuous potentials. When $d\geq 3$, the same result holds for frequencies under an explicit arithmetic criterion. We also show that general multi-dimensional operato… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.01782v1-abstract-full').style.display = 'inline'; document.getElementById('1712.01782v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1712.01782v1-abstract-full" style="display: none;"> We study the multi-dimensional operator $(H_x u)_n=\sum_{|m-n|=1}u_{m}+f(T^n(x))u_n$, where $T$ is the shift of the torus $\T^d$. When $d=2$, we show the spectrum of $H_x$ is almost surely purely continuous for a.e. $伪$ and generic continuous potentials. When $d\geq 3$, the same result holds for frequencies under an explicit arithmetic criterion. We also show that general multi-dimensional operators with measurable potentials do not have eigenvalue for generic $伪$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.01782v1-abstract-full').style.display = 'none'; document.getElementById('1712.01782v1-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 December, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1707.07259">arXiv:1707.07259</a> <span> [<a href="https://arxiv.org/pdf/1707.07259">pdf</a>, <a href="https://arxiv.org/format/1707.07259">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</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/10.1109/TPWRS.2018.2884876">10.1109/TPWRS.2018.2884876 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Hierarchical Plug-and-Play Voltage/Current Controller of DC Microgrid Clusters with Grid-Forming/Feeding Converters: Line-independent Primary Stabilization and Leader-based Distributed Secondary Regulation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Renke Han</a>, <a href="/search/math?searchtype=author&query=Tucci%2C+M">Michele Tucci</a>, <a href="/search/math?searchtype=author&query=Soloperto%2C+R">Raffaele Soloperto</a>, <a href="/search/math?searchtype=author&query=Martinelli%2C+A">Andrea Martinelli</a>, <a href="/search/math?searchtype=author&query=Ferrari-Trecate%2C+G">Giancarlo Ferrari-Trecate</a>, <a href="/search/math?searchtype=author&query=Guerrero%2C+J+M">Josep M. Guerrero</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="1707.07259v4-abstract-short" style="display: inline;"> Considering the structure of dc Microgrids (MGs) composed of grid-forming/feeding converters, a hierarchical Plug-and-Play (PnP) voltage/current control architecture for MG clusters is proposed. In the primary level, a PnP voltage/current controller is proposed to achieve simultaneous voltage support and current feeding function according to local references. In addition, stabilizing controller is… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.07259v4-abstract-full').style.display = 'inline'; document.getElementById('1707.07259v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1707.07259v4-abstract-full" style="display: none;"> Considering the structure of dc Microgrids (MGs) composed of grid-forming/feeding converters, a hierarchical Plug-and-Play (PnP) voltage/current control architecture for MG clusters is proposed. In the primary level, a PnP voltage/current controller is proposed to achieve simultaneous voltage support and current feeding function according to local references. In addition, stabilizing controller is characterized by explicit inequalities which are only related to local parameters of a MG. In the secondary level, for the system with interconnection of MGs, a leader-based voltage/current distributed controller is proposed to achieve both voltage and current regulation without specifying the individual setpoints for each MG. The proposed controller requires a communication network and each controller exchanges information with its communication neighbors only. With the proposed controller, each MG can plug-in/out of the system seamlessly, irrespectively of the power line parameters and models of other MGs . The proof of the MG cluster closed-loop stability exploits structured Lyapunov functions, the LaSalle invariance theorem and properties of graph Laplacians. Theoretical results are validated by hardware-in-loop (HiL) tests. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.07259v4-abstract-full').style.display = 'none'; document.getElementById('1707.07259v4-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> 19 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 July, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2017. </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">37 pages, 10 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> 2018 IEEE Transactions on Power Systems </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1707.03482">arXiv:1707.03482</a> <span> [<a href="https://arxiv.org/pdf/1707.03482">pdf</a>, <a href="https://arxiv.org/ps/1707.03482">ps</a>, <a href="https://arxiv.org/format/1707.03482">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey 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/10.1007/s00220-018-3141-9">10.1007/s00220-018-3141-9 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Discrete Bethe-Sommerfeld Conjecture </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Jitomirskaya%2C+S">Svetlana Jitomirskaya</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="1707.03482v1-abstract-short" style="display: inline;"> In this paper, we prove a discrete version of the Bethe-Sommerfeld conjecture. Namely, we show that the spectra of multi-dimensional discrete periodic Schr枚dinger operators on $\mathbb{Z}^d$ lattice with sufficiently small potentials contain at most two intervals. Moreover, the spectrum is a single interval, provided one of the periods is odd, and can have a gap whenever all periods are even. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1707.03482v1-abstract-full" style="display: none;"> In this paper, we prove a discrete version of the Bethe-Sommerfeld conjecture. Namely, we show that the spectra of multi-dimensional discrete periodic Schr枚dinger operators on $\mathbb{Z}^d$ lattice with sufficiently small potentials contain at most two intervals. Moreover, the spectrum is a single interval, provided one of the periods is odd, and can have a gap whenever all periods are even. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.03482v1-abstract-full').style.display = 'none'; document.getElementById('1707.03482v1-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 July, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2017. </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">10 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/1704.04603">arXiv:1704.04603</a> <span> [<a href="https://arxiv.org/pdf/1704.04603">pdf</a>, <a href="https://arxiv.org/ps/1704.04603">ps</a>, <a href="https://arxiv.org/format/1704.04603">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Shnol's theorem and the spectrum of long range operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</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="1704.04603v2-abstract-short" style="display: inline;"> We extend some basic results known for finite range operators to long range operators with off-diagonal decay. Namely, we prove an analogy of Sch'nol's theorem. We also establish the connection between the almost sure spectrum of long range random operators and the spectra of deterministic periodic operators. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1704.04603v2-abstract-full" style="display: none;"> We extend some basic results known for finite range operators to long range operators with off-diagonal decay. Namely, we prove an analogy of Sch'nol's theorem. We also establish the connection between the almost sure spectrum of long range random operators and the spectra of deterministic periodic operators. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1704.04603v2-abstract-full').style.display = 'none'; document.getElementById('1704.04603v2-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 September, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 April, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2017. </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">To appear in Proc. AMS. Referee's comments incorporated</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1608.01032">arXiv:1608.01032</a> <span> [<a href="https://arxiv.org/pdf/1608.01032">pdf</a>, <a href="https://arxiv.org/ps/1608.01032">ps</a>, <a href="https://arxiv.org/format/1608.01032">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</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.aim.2017.08.026">10.1016/j.aim.2017.08.026 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Full measure reducibility and localization for Jacobi operators: a topological criterion </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Jitomirskaya%2C+S">Svetlana Jitomirskaya</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="1608.01032v2-abstract-short" style="display: inline;"> We establish a topological criterion for connection between reducibility to constant rotations and dual localization, for the general family of analytic quasiperiodic Jacobi operators. As a corollary, we obtain the sharp arithmetic phase transition for the extended Harper's model in the positive Lyapunov exponent region. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1608.01032v2-abstract-full" style="display: none;"> We establish a topological criterion for connection between reducibility to constant rotations and dual localization, for the general family of analytic quasiperiodic Jacobi operators. As a corollary, we obtain the sharp arithmetic phase transition for the extended Harper's model in the positive Lyapunov exponent region. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1608.01032v2-abstract-full').style.display = 'none'; document.getElementById('1608.01032v2-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 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 August, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2016. </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">Referee comments incorporated</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 47B36 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Advances in Mathematics 319C (2017) pp. 224-250 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1607.08576">arXiv:1607.08576</a> <span> [<a href="https://arxiv.org/pdf/1607.08576">pdf</a>, <a href="https://arxiv.org/ps/1607.08576">ps</a>, <a href="https://arxiv.org/format/1607.08576">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</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/10.2140/apde.2019.12.867">10.2140/apde.2019.12.867 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantum dynamical bounds for ergodic potentials with underlying dynamics of zero topological entropy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</a>, <a href="/search/math?searchtype=author&query=Jitomirskaya%2C+S">Svetlana Jitomirskaya</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="1607.08576v1-abstract-short" style="display: inline;"> In this paper we obtain upper quantum dynamical bounds as a corollary of positive Lyapunov exponent for Schr枚dinger operators $H_{f,胃} u(n)=u(n+1)+u(n-1)+ 蠁(f^n胃)u(n)$, where $蠁: \mathcal{M}\to {\Bbb R}$ is a piecewise H枚lder function on a compact Riemannian manifold $\mathcal{M}$, and $f:\mathcal{M}\to\mathcal{M}$ is a uniquely ergodic volume preserving map with zero topological entropy. As corol… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1607.08576v1-abstract-full').style.display = 'inline'; document.getElementById('1607.08576v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1607.08576v1-abstract-full" style="display: none;"> In this paper we obtain upper quantum dynamical bounds as a corollary of positive Lyapunov exponent for Schr枚dinger operators $H_{f,胃} u(n)=u(n+1)+u(n-1)+ 蠁(f^n胃)u(n)$, where $蠁: \mathcal{M}\to {\Bbb R}$ is a piecewise H枚lder function on a compact Riemannian manifold $\mathcal{M}$, and $f:\mathcal{M}\to\mathcal{M}$ is a uniquely ergodic volume preserving map with zero topological entropy. As corollaries we obtain localization-type statements for shifts and skew-shifts on higher dimensional tori with arithmetic conditions on the parameters. These are the first localization-type results with precise arithmetic conditions for multi-frequency quasiperiodic and skew-shift potentials. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1607.08576v1-abstract-full').style.display = 'none'; document.getElementById('1607.08576v1-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 July, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2016. </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> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 47B36 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Analysis & PDE 12 (2019) 867-902 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1607.08571">arXiv:1607.08571</a> <span> [<a href="https://arxiv.org/pdf/1607.08571">pdf</a>, <a href="https://arxiv.org/ps/1607.08571">ps</a>, <a href="https://arxiv.org/format/1607.08571">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Dry Ten Martini problem for non self-dual extended Harper's model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</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="1607.08571v1-abstract-short" style="display: inline;"> In this paper we prove the dry version of the Ten Martini problem: Cantor spectrum with all gaps open, for the extended Harper's model in the non self-dual region for Diophantine frequencies. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1607.08571v1-abstract-full" style="display: none;"> In this paper we prove the dry version of the Ten Martini problem: Cantor spectrum with all gaps open, for the extended Harper's model in the non self-dual region for Diophantine frequencies. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1607.08571v1-abstract-full').style.display = 'none'; document.getElementById('1607.08571v1-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 July, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2016. </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">To appear in Transactions of the American Mathematical Society</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 47B36 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1607.08566">arXiv:1607.08566</a> <span> [<a href="https://arxiv.org/pdf/1607.08566">pdf</a>, <a href="https://arxiv.org/ps/1607.08566">ps</a>, <a href="https://arxiv.org/format/1607.08566">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Uniform localization is always uniform </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Rui Han</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="1607.08566v1-abstract-short" style="display: inline;"> In this note we show that if a family of ergodic Schr枚dinger operators on $l^2({\Bbb Z}^纬)$ with continuous potentials have uniformly localized eigenfunctions then these eigenfunctions must be uniformly localized in a homogeneous sense. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1607.08566v1-abstract-full" style="display: none;"> In this note we show that if a family of ergodic Schr枚dinger operators on $l^2({\Bbb Z}^纬)$ with continuous potentials have uniformly localized eigenfunctions then these eigenfunctions must be uniformly localized in a homogeneous sense. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1607.08566v1-abstract-full').style.display = 'none'; document.getElementById('1607.08566v1-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 July, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 47B36 (Primary); 81Q10 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Proc. Amer. Math. 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