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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=Yang%2C+Y&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Yang%2C+Y&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Yang%2C+Y&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Yang%2C+Y&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> <li> <a href="/search/?searchtype=author&query=Yang%2C+Y&start=150" class="pagination-link " aria-label="Page 4" aria-current="page">4 </a> </li> <li> <a href="/search/?searchtype=author&query=Yang%2C+Y&start=200" class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </li> <li><span class="pagination-ellipsis">…</span></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/2411.06393">arXiv:2411.06393</a> <span> [<a href="https://arxiv.org/pdf/2411.06393">pdf</a>, <a href="https://arxiv.org/format/2411.06393">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> <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"> Evolution of weights on a connected finite graph </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jicheng Ma</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yunyan 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="2411.06393v1-abstract-short" style="display: inline;"> On a connected finite graph, we propose an evolution of weights including Ollivier's Ricci flow as a special case. During the evolution process, on each edge, the speed of change of weight is exactly the difference between the Wasserstein distance related to two probability measures and certain graph distance. Here the probability measure may be chosen as an $伪$-lazy one-step random walk, an $伪$-l… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06393v1-abstract-full').style.display = 'inline'; document.getElementById('2411.06393v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.06393v1-abstract-full" style="display: none;"> On a connected finite graph, we propose an evolution of weights including Ollivier's Ricci flow as a special case. During the evolution process, on each edge, the speed of change of weight is exactly the difference between the Wasserstein distance related to two probability measures and certain graph distance. Here the probability measure may be chosen as an $伪$-lazy one-step random walk, an $伪$-lazy two-step random walk, or a general probability measure. Based on the ODE theory, we show that the initial value problem has a unique global solution. A discrete version of the above evolution is applied to the problem of community detection. Our algorithm is based on such a discrete evolution, where probability measures are chosen as $伪$-lazy one-step random walk and $伪$-lazy two-step random walk respectively. Note that the later measure has not been used in previous works [2, 16, 20, 23]. Here, as in [20], only one surgery needs to be performed after the last iteration. Moreover, our algorithm is much easier than those of [2, 16, 20], which were all based on Lin-Lu-Yau's Ricci curvature. The code is available at https://github.com/mjc191812/Evolution-of-weights-on-a-connected-finite-graph. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06393v1-abstract-full').style.display = 'none'; document.getElementById('2411.06393v1-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 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">35 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C21; 05C85; 35R02; 68Q06 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.05332">arXiv:2411.05332</a> <span> [<a href="https://arxiv.org/pdf/2411.05332">pdf</a>, <a href="https://arxiv.org/ps/2411.05332">ps</a>, <a href="https://arxiv.org/format/2411.05332">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Sparse Principal Component Analysis with Non-Oblivious Adversarial Perturbations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=He%2C+Y">Yuqing He</a>, <a href="/search/math?searchtype=author&query=Wang%2C+G">Guanyi Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yu 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="2411.05332v1-abstract-short" style="display: inline;"> Sparse Principal Component Analysis (sparse PCA) is a fundamental dimension-reduction tool that enhances interpretability in various high-dimensional settings. An important variant of sparse PCA studies the scenario when samples are adversarially perturbed. Notably, most existing statistical studies on this variant focus on recovering the ground truth and verifying the robustness of classical algo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.05332v1-abstract-full').style.display = 'inline'; document.getElementById('2411.05332v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.05332v1-abstract-full" style="display: none;"> Sparse Principal Component Analysis (sparse PCA) is a fundamental dimension-reduction tool that enhances interpretability in various high-dimensional settings. An important variant of sparse PCA studies the scenario when samples are adversarially perturbed. Notably, most existing statistical studies on this variant focus on recovering the ground truth and verifying the robustness of classical algorithms when the given samples are corrupted under oblivious adversarial perturbations. In contrast, this paper aims to find a robust sparse principal component that maximizes the variance of the given samples corrupted by non-oblivious adversarial perturbations, say sparse PCA with Non-Oblivious Adversarial Perturbations (sparse PCA-NOAP). Specifically, we introduce a general formulation for the proposed sparse PCA-NOAP. We then derive Mixed-Integer Programming (MIP) reformulations to upper bound it with provable worst-case guarantees when adversarial perturbations are controlled by two typical norms, i.e., $\ell_{2 \rightarrow \infty}$-norm (sample-wise $\ell_2$-norm perturbation) and $\ell_{1 \rightarrow 2}$-norm (feature-wise $\ell_2$-norm perturbation). Moreover, when samples are drawn from the spiked Wishart model, we show that the proposed MIP reformulations ensure vector recovery properties under a more general parameter region compared with existing results. Numerical simulations are also provided to validate the theoretical findings and demonstrate the accuracy of the proposed formulations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.05332v1-abstract-full').style.display = 'none'; document.getElementById('2411.05332v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.01237">arXiv:2411.01237</a> <span> [<a href="https://arxiv.org/pdf/2411.01237">pdf</a>, <a href="https://arxiv.org/ps/2411.01237">ps</a>, <a href="https://arxiv.org/format/2411.01237">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"> Sparse Linear Regression: Sequential Convex Relaxation, Robust Restricted Null Space Property, and Variable Selection </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bi%2C+S">Shujun Bi</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yonghua Yang</a>, <a href="/search/math?searchtype=author&query=Pan%2C+S">Shaohua Pan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.01237v1-abstract-short" style="display: inline;"> For high dimensional sparse linear regression problems, we propose a sequential convex relaxation algorithm (iSCRA-TL1) by solving inexactly a sequence of truncated $\ell_1$-norm regularized minimization problems, in which the working index sets are constructed iteratively with an adaptive strategy. We employ the robust restricted null space property and sequential restricted null space property (… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.01237v1-abstract-full').style.display = 'inline'; document.getElementById('2411.01237v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.01237v1-abstract-full" style="display: none;"> For high dimensional sparse linear regression problems, we propose a sequential convex relaxation algorithm (iSCRA-TL1) by solving inexactly a sequence of truncated $\ell_1$-norm regularized minimization problems, in which the working index sets are constructed iteratively with an adaptive strategy. We employ the robust restricted null space property and sequential restricted null space property (rRNSP and rSRNSP) to provide the theoretical certificates of iSCRA-TL1. Specifically, under a mild rRNSP or rSRNSP, iSCRA-TL1 is shown to identify the support of the true $r$-sparse vector by solving at most $r$ truncated $\ell_1$-norm regularized problems, and the $\ell_1$-norm error bound of its iterates from the oracle solution is also established. As a consequence, an oracle estimator of high-dimensional linear regression problems can be achieved by solving at most $r\!+\!1$ truncated $\ell_1$-norm regularized problems. To the best of our knowledge, this is the first sequential convex relaxation algorithm to produce an oracle estimator under a weaker NSP condition within a specific number of steps, provided that the Lasso estimator lacks high quality, say, the supports of its first $r$ largest (in modulus) entries do not coincide with those of the true vector. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.01237v1-abstract-full').style.display = 'none'; document.getElementById('2411.01237v1-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 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">38 pages, 4 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/2410.23328">arXiv:2410.23328</a> <span> [<a href="https://arxiv.org/pdf/2410.23328">pdf</a>, <a href="https://arxiv.org/ps/2410.23328">ps</a>, <a href="https://arxiv.org/format/2410.23328">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Higher Dimensional Versions of the Douglas-Ahlfors Identities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yan Yang</a>, <a href="/search/math?searchtype=author&query=Qian%2C+T">Tao Qian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.23328v4-abstract-short" style="display: inline;"> Denote by ${\mathcal D}$ the open unit disc in the complex plane and $\partial {\mathcal D}$ its boundary. Douglas showed through an identical quantity represented by the Fourier coefficients of the concerned function $u$ that \begin{eqnarray}\label{abs} A(u)=\int_{\mathcal D}|\bigtriangledown U|^2dxdy&=&\frac{1}{2蟺}\int\int_{\partial {\mathcal D}\times \partial {\mathcal D}} \left|\frac{u(z_1)-u(… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.23328v4-abstract-full').style.display = 'inline'; document.getElementById('2410.23328v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.23328v4-abstract-full" style="display: none;"> Denote by ${\mathcal D}$ the open unit disc in the complex plane and $\partial {\mathcal D}$ its boundary. Douglas showed through an identical quantity represented by the Fourier coefficients of the concerned function $u$ that \begin{eqnarray}\label{abs} A(u)=\int_{\mathcal D}|\bigtriangledown U|^2dxdy&=&\frac{1}{2蟺}\int\int_{\partial {\mathcal D}\times \partial {\mathcal D}} \left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|^2|dz_1||dz_2|,\end{eqnarray} \end{abstract} where $u\in L^2(\partial {\mathcal D}), U$ is the harmonic extension of $u$ into ${\mathcal D}$. Ahlfors gave a fourth equivalence form of $A(u)$ in (\ref{more}) via a different proof. The present article studies relations between the counterpart quantities in higher dimensional spheres with several different but commonly adopted settings, namely, harmonic functions in the Euclidean ${\mathbb R}^n, n\ge 2,$ regular functions in the quaternionic algebra, and Clifford monogenic functions with the real-Clifford algebra ${\mathcal{CL}}_{0, n-1},$ the latter being generated by the multiplication anti-commutative basic imaginary units ${\e}_1, {\e}_2, \cdots , {\e}_{n-1}$ with ${\e}_j^2=-1, j=1, 2, \cdots, n-1.$ It is noted that, while exactly the same equivalence relations hold for harmonic functions in ${\mathbb R}^n$ and regular functions in the quaternionic algebra, for the Clifford algebra setting $n>2,$ the relation (\ref{more}) has to be replaced by essentially a different rule. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.23328v4-abstract-full').style.display = 'none'; document.getElementById('2410.23328v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.20489">arXiv:2410.20489</a> <span> [<a href="https://arxiv.org/pdf/2410.20489">pdf</a>, <a href="https://arxiv.org/format/2410.20489">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> On the Volume Conjecture for hyperbolic Dehn-filled 3-manifolds along the twist knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ge%2C+H">Huabin Ge</a>, <a href="/search/math?searchtype=author&query=Meng%2C+Y">Yunpeng Meng</a>, <a href="/search/math?searchtype=author&query=Wang%2C+C">Chuwen Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuxuan 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="2410.20489v1-abstract-short" style="display: inline;"> For a twist knot $\mathcal{K}_{p'}$, let $M$ be the closed $3$-manifold obtained by doing $(p, q)$ Dehn-filling along $\mathcal{K}_{p'}$. In this article, we prove that Chen-Yang's volume conjecture holds for sufficiently large $|p| + |q|$ and $|p'|$ for $M$. In the proof, we construct a new ideal triangulation of the Whitehead link complement which is different from Thurston's triangulation. Our… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.20489v1-abstract-full').style.display = 'inline'; document.getElementById('2410.20489v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.20489v1-abstract-full" style="display: none;"> For a twist knot $\mathcal{K}_{p'}$, let $M$ be the closed $3$-manifold obtained by doing $(p, q)$ Dehn-filling along $\mathcal{K}_{p'}$. In this article, we prove that Chen-Yang's volume conjecture holds for sufficiently large $|p| + |q|$ and $|p'|$ for $M$. In the proof, we construct a new ideal triangulation of the Whitehead link complement which is different from Thurston's triangulation. Our triangulation has led to some new discoveries regarding symmetry, including insights into ``sister manifolds'' as introduced by Hodgson, Meyerhoff, and Weeks. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.20489v1-abstract-full').style.display = 'none'; document.getElementById('2410.20489v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">58 pages, 34 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/2410.18439">arXiv:2410.18439</a> <span> [<a href="https://arxiv.org/pdf/2410.18439">pdf</a>, <a href="https://arxiv.org/format/2410.18439">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"> Multiscale Neural Networks for Approximating Green's Functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hao%2C+W">Wenrui Hao</a>, <a href="/search/math?searchtype=author&query=Li%2C+R+P">Rui Peng Li</a>, <a href="/search/math?searchtype=author&query=Xi%2C+Y">Yuanzhe Xi</a>, <a href="/search/math?searchtype=author&query=Xu%2C+T">Tianshi Xu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yahong 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="2410.18439v2-abstract-short" style="display: inline;"> Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green's functions. However, Green's functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer traini… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.18439v2-abstract-full').style.display = 'inline'; document.getElementById('2410.18439v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.18439v2-abstract-full" style="display: none;"> Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green's functions. However, Green's functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer training times. In this paper, we address these challenges by leveraging multiscale NNs to learn Green's functions. Through theoretical analysis using multiscale Barron space methods and experimental validation, we show that the multiscale approach significantly reduces the necessary NN size and accelerates training. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.18439v2-abstract-full').style.display = 'none'; document.getElementById('2410.18439v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">21 pages, 11 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65N55; 65N80; 68T07 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.16281">arXiv:2410.16281</a> <span> [<a href="https://arxiv.org/pdf/2410.16281">pdf</a>, <a href="https://arxiv.org/format/2410.16281">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Robotics">cs.RO</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="Systems and Control">eess.SY</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"> Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hu%2C+H">Hanjiang Hu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yujie Yang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+T">Tianhao Wei</a>, <a href="/search/math?searchtype=author&query=Liu%2C+C">Changliu Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.16281v1-abstract-short" style="display: inline;"> Control barrier functions (CBFs) are important in safety-critical systems and robot control applications. Neural networks have been used to parameterize and synthesize CBFs with bounded control input for complex systems. However, it is still challenging to verify pre-trained neural networks CBFs (neural CBFs) in an efficient symbolic manner. To this end, we propose a new efficient verification fra… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.16281v1-abstract-full').style.display = 'inline'; document.getElementById('2410.16281v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.16281v1-abstract-full" style="display: none;"> Control barrier functions (CBFs) are important in safety-critical systems and robot control applications. Neural networks have been used to parameterize and synthesize CBFs with bounded control input for complex systems. However, it is still challenging to verify pre-trained neural networks CBFs (neural CBFs) in an efficient symbolic manner. To this end, we propose a new efficient verification framework for ReLU-based neural CBFs through symbolic derivative bound propagation by combining the linearly bounded nonlinear dynamic system and the gradient bounds of neural CBFs. Specifically, with Heaviside step function form for derivatives of activation functions, we show that the symbolic bounds can be propagated through the inner product of neural CBF Jacobian and nonlinear system dynamics. Through extensive experiments on different robot dynamics, our results outperform the interval arithmetic based baselines in verified rate and verification time along the CBF boundary, validating the effectiveness and efficiency of the proposed method with different model complexity. The code can be found at https://github.com/intelligent-control-lab/ verify-neural-CBF. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.16281v1-abstract-full').style.display = 'none'; document.getElementById('2410.16281v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">Accepted to CoRL 2024, 18 pages, 6 figures, 4 tables</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.15331">arXiv:2410.15331</a> <span> [<a href="https://arxiv.org/pdf/2410.15331">pdf</a>, <a href="https://arxiv.org/format/2410.15331">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 novel polyhedral scaled boundary finite element method solving three-dimensional heat conduction problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yan%2C+M">Mingjiao Yan</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yang Yang</a>, <a href="/search/math?searchtype=author&query=Su%2C+C">Chao Su</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Z">Zongliang Zhang</a>, <a href="/search/math?searchtype=author&query=Duan%2C+Q">Qingsong Duan</a>, <a href="/search/math?searchtype=author&query=Hao%2C+D">Dengmiao Hao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.15331v2-abstract-short" style="display: inline;"> In this work, we derived the three-dimensional scaled boundary finite element formulation for thermal conduction problems. By introducing Wachspress shape functions, we proposed a novel polyhedral scaled boundary finite element method (PSBFEM) to address thermal conduction problems. The proposed method effectively addresses the challenges associated with complex geometries by integrating the polyh… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.15331v2-abstract-full').style.display = 'inline'; document.getElementById('2410.15331v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.15331v2-abstract-full" style="display: none;"> In this work, we derived the three-dimensional scaled boundary finite element formulation for thermal conduction problems. By introducing Wachspress shape functions, we proposed a novel polyhedral scaled boundary finite element method (PSBFEM) to address thermal conduction problems. The proposed method effectively addresses the challenges associated with complex geometries by integrating the polyhedral mesh and the octree mesh. The presented formulation handles both steady-state and transient thermal conduction analyses. Through a series of numerical examples, the accuracy and convergence of the proposed method were validated. The results demonstrate that mesh refinement leads to superior accuracy for the PSBFEM compared to the FEM. Moreover, Polyhedral elements provide an effective and efficient approach for complex simulations that substantially reduces computational costs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.15331v2-abstract-full').style.display = 'none'; document.getElementById('2410.15331v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.14346">arXiv:2410.14346</a> <span> [<a href="https://arxiv.org/pdf/2410.14346">pdf</a>, <a href="https://arxiv.org/ps/2410.14346">ps</a>, <a href="https://arxiv.org/format/2410.14346">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> Conformal weldings in the Loewner equation and Weil--Petersson quasislit-disks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Tao%2C+F">Fei Tao</a>, <a href="/search/math?searchtype=author&query=Wei%2C+H">Huaying Wei</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yaosong 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="2410.14346v1-abstract-short" style="display: inline;"> A simple arc $螕= 纬(0, T]$, growing into the unit disk $\mathbb D$ from its boundary, generates a driving term $尉$ and a conformal welding $蠁$ through the Loewner differential equation. When $螕$ is the slit of a Weil--Petersson quasislit-disk $\mathbb D\setminus螕$, the Loewner transform and its inverse $螕\leftrightarrow 尉$ have been well understood due to Y. Wang's work. We investigate the maps… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.14346v1-abstract-full').style.display = 'inline'; document.getElementById('2410.14346v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.14346v1-abstract-full" style="display: none;"> A simple arc $螕= 纬(0, T]$, growing into the unit disk $\mathbb D$ from its boundary, generates a driving term $尉$ and a conformal welding $蠁$ through the Loewner differential equation. When $螕$ is the slit of a Weil--Petersson quasislit-disk $\mathbb D\setminus螕$, the Loewner transform and its inverse $螕\leftrightarrow 尉$ have been well understood due to Y. Wang's work. We investigate the maps $螕\leftrightarrow 蠁$ in this case, giving a description of $螕$ in terms of $蠁$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.14346v1-abstract-full').style.display = 'none'; document.getElementById('2410.14346v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">12 pages, 3 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 30C62; 30C75; 30H25 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.14054">arXiv:2410.14054</a> <span> [<a href="https://arxiv.org/pdf/2410.14054">pdf</a>, <a href="https://arxiv.org/format/2410.14054">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="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Independently-Normalized SGD for Generalized-Smooth Nonconvex Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yufeng Yang</a>, <a href="/search/math?searchtype=author&query=Tripp%2C+E">Erin Tripp</a>, <a href="/search/math?searchtype=author&query=Sun%2C+Y">Yifan Sun</a>, <a href="/search/math?searchtype=author&query=Zou%2C+S">Shaofeng Zou</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+Y">Yi 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="2410.14054v1-abstract-short" style="display: inline;"> Recent studies have shown that many nonconvex machine learning problems meet a so-called generalized-smooth condition that extends beyond traditional smooth nonconvex optimization. However, the existing algorithms designed for generalized-smooth nonconvex optimization encounter significant limitations in both their design and convergence analysis. In this work, we first study deterministic general… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.14054v1-abstract-full').style.display = 'inline'; document.getElementById('2410.14054v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.14054v1-abstract-full" style="display: none;"> Recent studies have shown that many nonconvex machine learning problems meet a so-called generalized-smooth condition that extends beyond traditional smooth nonconvex optimization. However, the existing algorithms designed for generalized-smooth nonconvex optimization encounter significant limitations in both their design and convergence analysis. In this work, we first study deterministic generalized-smooth nonconvex optimization and analyze the convergence of normalized gradient descent under the generalized Polyak-Lojasiewicz condition. Our results provide a comprehensive understanding of the interplay between gradient normalization and function geometry. Then, for stochastic generalized-smooth nonconvex optimization, we propose an independently-normalized stochastic gradient descent algorithm, which leverages independent sampling, gradient normalization and clipping to achieve an $\mathcal{O}(蔚^{-4})$ sample complexity under relaxed assumptions. Experiments demonstrate the fast convergence of our algorithm. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.14054v1-abstract-full').style.display = 'none'; document.getElementById('2410.14054v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <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">3 figures, 30 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.11382">arXiv:2410.11382</a> <span> [<a href="https://arxiv.org/pdf/2410.11382">pdf</a>, <a href="https://arxiv.org/format/2410.11382">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Point-Calibrated Spectral Neural Operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yue%2C+X">Xihang Yue</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+L">Linchao Zhu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yi 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="2410.11382v1-abstract-short" style="display: inline;"> Two typical neural models have been extensively studied for operator learning, learning in spatial space via attention mechanism or learning in spectral space via spectral analysis technique such as Fourier Transform. Spatial learning enables point-level flexibility but lacks global continuity constraint, while spectral learning enforces spectral continuity prior but lacks point-wise adaptivity. T… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.11382v1-abstract-full').style.display = 'inline'; document.getElementById('2410.11382v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.11382v1-abstract-full" style="display: none;"> Two typical neural models have been extensively studied for operator learning, learning in spatial space via attention mechanism or learning in spectral space via spectral analysis technique such as Fourier Transform. Spatial learning enables point-level flexibility but lacks global continuity constraint, while spectral learning enforces spectral continuity prior but lacks point-wise adaptivity. This work innovatively combines the continuity prior and the point-level flexibility, with the introduced Point-Calibrated Spectral Transform. It achieves this by calibrating the preset spectral eigenfunctions with the predicted point-wise frequency preference via neural gate mechanism. Beyond this, we introduce Point-Calibrated Spectral Neural Operators, which learn operator mappings by approximating functions with the point-level adaptive spectral basis, thereby not only preserving the benefits of spectral prior but also boasting the superior adaptability comparable to the attention mechanism. Comprehensive experiments demonstrate its consistent performance enhancement in extensive PDE solving scenarios. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.11382v1-abstract-full').style.display = 'none'; document.getElementById('2410.11382v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.09969">arXiv:2410.09969</a> <span> [<a href="https://arxiv.org/pdf/2410.09969">pdf</a>, <a href="https://arxiv.org/ps/2410.09969">ps</a>, <a href="https://arxiv.org/format/2410.09969">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</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"> Remarks on $p$-primary torsion of the Brauer group </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuan 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="2410.09969v1-abstract-short" style="display: inline;"> For a smooth and proper variety $X$ over an algebraically closed field $k$ of characteristic $p>0$, the group $Br(X)[p^\infty]$ is a direct sum of finitely many copies of $\mathbb{Q}_p/\mathbb{Z}_p$ and an abelian group of finite exponent. The latter is an extension of a finite group $J$ by the group of $k$-points of a connected commutative unipotent algebraic group $U$. In this paper we show that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09969v1-abstract-full').style.display = 'inline'; document.getElementById('2410.09969v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.09969v1-abstract-full" style="display: none;"> For a smooth and proper variety $X$ over an algebraically closed field $k$ of characteristic $p>0$, the group $Br(X)[p^\infty]$ is a direct sum of finitely many copies of $\mathbb{Q}_p/\mathbb{Z}_p$ and an abelian group of finite exponent. The latter is an extension of a finite group $J$ by the group of $k$-points of a connected commutative unipotent algebraic group $U$. In this paper we show that (1) if $X$ is ordinary, then $U = 0$; (2) if $X$ is a surface, then $J$ is the Pontryagin dual of $NS(X)[p^\infty]$; (3) if $X$ is an abelian variety, then $J = 0$. Using Crew's formula, we compute $Br(X)[p^\infty]$ for Enriques surfaces and abelian $3$-folds. Generalizing a result of Ogus, we give a criterion for the injectivity of the canonical map from flat to crystalline cohomology in degree $2$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09969v1-abstract-full').style.display = 'none'; document.getElementById('2410.09969v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.09402">arXiv:2410.09402</a> <span> [<a href="https://arxiv.org/pdf/2410.09402">pdf</a>, <a href="https://arxiv.org/ps/2410.09402">ps</a>, <a href="https://arxiv.org/format/2410.09402">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"> Minimax rates of convergence for nonparametric regression under adversarial attacks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Peng%2C+J">Jingfu Peng</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuhong 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="2410.09402v1-abstract-short" style="display: inline;"> Recent research shows the susceptibility of machine learning models to adversarial attacks, wherein minor but maliciously chosen perturbations of the input can significantly degrade model performance. In this paper, we theoretically analyse the limits of robustness against such adversarial attacks in a nonparametric regression setting, by examining the minimax rates of convergence in an adversaria… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09402v1-abstract-full').style.display = 'inline'; document.getElementById('2410.09402v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.09402v1-abstract-full" style="display: none;"> Recent research shows the susceptibility of machine learning models to adversarial attacks, wherein minor but maliciously chosen perturbations of the input can significantly degrade model performance. In this paper, we theoretically analyse the limits of robustness against such adversarial attacks in a nonparametric regression setting, by examining the minimax rates of convergence in an adversarial sup-norm. Our work reveals that the minimax rate under adversarial attacks in the input is the same as sum of two terms: one represents the minimax rate in the standard setting without adversarial attacks, and the other reflects the maximum deviation of the true regression function value within the target function class when subjected to the input perturbations. The optimal rates under the adversarial setup can be achieved by a plug-in procedure constructed from a minimax optimal estimator in the corresponding standard setting. Two specific examples are given to illustrate the established minimax results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09402v1-abstract-full').style.display = 'none'; document.getElementById('2410.09402v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.07556">arXiv:2410.07556</a> <span> [<a href="https://arxiv.org/pdf/2410.07556">pdf</a>, <a href="https://arxiv.org/ps/2410.07556">ps</a>, <a href="https://arxiv.org/format/2410.07556">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> On The Largest Character Degree And Solvable Subgroups Of Finite Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+Z">Zongshu Wu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yong 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="2410.07556v2-abstract-short" style="display: inline;"> Let $G$ be a finite group, and $蟺$ be a set of primes. The $蟺$-core $\mathbf{O}_蟺(G)$ is the unique maximal normal $蟺$-subgroup of $G$, and $b(G)$ is the largest irreducible character degree of $G$. In 2017, Qian and Yang proved that if $H$ is a solvable $蟺$-subgroup of $G$, then $|H\mathbf{O}_蟺(G)/\mathbf{O}_蟺(G)|\le b(G)^3$. In this paper, we improve the exponent of $3$ to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.07556v2-abstract-full').style.display = 'inline'; document.getElementById('2410.07556v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.07556v2-abstract-full" style="display: none;"> Let $G$ be a finite group, and $蟺$ be a set of primes. The $蟺$-core $\mathbf{O}_蟺(G)$ is the unique maximal normal $蟺$-subgroup of $G$, and $b(G)$ is the largest irreducible character degree of $G$. In 2017, Qian and Yang proved that if $H$ is a solvable $蟺$-subgroup of $G$, then $|H\mathbf{O}_蟺(G)/\mathbf{O}_蟺(G)|\le b(G)^3$. In this paper, we improve the exponent of $3$ to $3\log_{504}(168)<2.471$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.07556v2-abstract-full').style.display = 'none'; document.getElementById('2410.07556v2-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, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.06486">arXiv:2410.06486</a> <span> [<a href="https://arxiv.org/pdf/2410.06486">pdf</a>, <a href="https://arxiv.org/ps/2410.06486">ps</a>, <a href="https://arxiv.org/format/2410.06486">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Outer Independent Roman Domination Number of Cartesian Product of Paths and Cycles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+H">Hong Gao</a>, <a href="/search/math?searchtype=author&query=Qiu%2C+D">Daoda Qiu</a>, <a href="/search/math?searchtype=author&query=Du%2C+S">Shuyan Du</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+Y">Yiyue Zhao</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuansheng 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="2410.06486v1-abstract-short" style="display: inline;"> Given a graph $G$ with vertex set $V$, an outer independent Roman dominating function (OIRDF) is a function $f$ from $V(G)$ to $\{0, 1, 2\}$ for which every vertex with label $0$ under $f$ is adjacent to at least a vertex with label $2$ but not adjacent to another vertex with label $0$. The weight of an OIRDF $f$ is the sum of vertex function values all over the graph, and the minimum of an OIRDF… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06486v1-abstract-full').style.display = 'inline'; document.getElementById('2410.06486v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.06486v1-abstract-full" style="display: none;"> Given a graph $G$ with vertex set $V$, an outer independent Roman dominating function (OIRDF) is a function $f$ from $V(G)$ to $\{0, 1, 2\}$ for which every vertex with label $0$ under $f$ is adjacent to at least a vertex with label $2$ but not adjacent to another vertex with label $0$. The weight of an OIRDF $f$ is the sum of vertex function values all over the graph, and the minimum of an OIRDF is the outer independent Roman domination number of $G$, denoted as $纬_{oiR}(G)$. In this paper, we focus on the outer independent Roman domination number of the Cartesian product of paths and cycles $P_{n}\Box C_{m}$. We determine the exact values of $纬_{oiR}(P_n\Box C_m)$ for $n=1,2,3$ and $纬_{oiR}(P_n\Box C_3)$ and present an upper bound of $纬_{oiR}(P_n\Box C_m)$ for $n\ge 4, m\ge 4$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06486v1-abstract-full').style.display = 'none'; document.getElementById('2410.06486v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.06308">arXiv:2410.06308</a> <span> [<a href="https://arxiv.org/pdf/2410.06308">pdf</a>, <a href="https://arxiv.org/format/2410.06308">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Quantifying Training Difficulty and Accelerating Convergence in Neural Network-Based PDE Solvers </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+C">Chuqi Chen</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+Q">Qixuan Zhou</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yahong Yang</a>, <a href="/search/math?searchtype=author&query=Xiang%2C+Y">Yang Xiang</a>, <a href="/search/math?searchtype=author&query=Luo%2C+T">Tao Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.06308v1-abstract-short" style="display: inline;"> Neural network-based methods have emerged as powerful tools for solving partial differential equations (PDEs) in scientific and engineering applications, particularly when handling complex domains or incorporating empirical data. These methods leverage neural networks as basis functions to approximate PDE solutions. However, training such networks can be challenging, often resulting in limited acc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06308v1-abstract-full').style.display = 'inline'; document.getElementById('2410.06308v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.06308v1-abstract-full" style="display: none;"> Neural network-based methods have emerged as powerful tools for solving partial differential equations (PDEs) in scientific and engineering applications, particularly when handling complex domains or incorporating empirical data. These methods leverage neural networks as basis functions to approximate PDE solutions. However, training such networks can be challenging, often resulting in limited accuracy. In this paper, we investigate the training dynamics of neural network-based PDE solvers with a focus on the impact of initialization techniques. We assess training difficulty by analyzing the eigenvalue distribution of the kernel and apply the concept of effective rank to quantify this difficulty, where a larger effective rank correlates with faster convergence of the training error. Building upon this, we discover through theoretical analysis and numerical experiments that two initialization techniques, partition of unity (PoU) and variance scaling (VS), enhance the effective rank, thereby accelerating the convergence of training error. Furthermore, comprehensive experiments using popular PDE-solving frameworks, such as PINN, Deep Ritz, and the operator learning framework DeepOnet, confirm that these initialization techniques consistently speed up convergence, in line with our theoretical findings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06308v1-abstract-full').style.display = 'none'; document.getElementById('2410.06308v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.05847">arXiv:2410.05847</a> <span> [<a href="https://arxiv.org/pdf/2410.05847">pdf</a>, <a href="https://arxiv.org/ps/2410.05847">ps</a>, <a href="https://arxiv.org/format/2410.05847">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Forward self-similar solutions of the MHD-boussinesq system with newtonian gravitational field </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yifan 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="2410.05847v1-abstract-short" style="display: inline;"> This paper is concerned with the existence of forward self-similar solutions to the three-dimensional Magnetohydrodynamic-Boussinesq (MHD-Boussinesq) system with newtonian gravitational field. By employing a blow-up argument and the Leray-Schauder theorem, we construct a forward self-similar solution to this system without imposing any smallness assumptions on the self-similar initial value. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.05847v1-abstract-full" style="display: none;"> This paper is concerned with the existence of forward self-similar solutions to the three-dimensional Magnetohydrodynamic-Boussinesq (MHD-Boussinesq) system with newtonian gravitational field. By employing a blow-up argument and the Leray-Schauder theorem, we construct a forward self-similar solution to this system without imposing any smallness assumptions on the self-similar initial value. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.05847v1-abstract-full').style.display = 'none'; document.getElementById('2410.05847v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.05486">arXiv:2410.05486</a> <span> [<a href="https://arxiv.org/pdf/2410.05486">pdf</a>, <a href="https://arxiv.org/format/2410.05486">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Multi-Window Approaches for Direct and Stable STFT Phase Retrieval </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Alaifari%2C+R">Rima Alaifari</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="2410.05486v1-abstract-short" style="display: inline;"> Phase retrieval from phaseless short-time Fourier transform (STFT) measurements is known to be inherently unstable when measurements are taken with respect to a single window. While an explicit inversion formula exists, it is useless in practice due to its instability. In this paper, we overcome this lack of stability by presenting two multi-window approaches that rely on a "good coverage" of the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.05486v1-abstract-full').style.display = 'inline'; document.getElementById('2410.05486v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.05486v1-abstract-full" style="display: none;"> Phase retrieval from phaseless short-time Fourier transform (STFT) measurements is known to be inherently unstable when measurements are taken with respect to a single window. While an explicit inversion formula exists, it is useless in practice due to its instability. In this paper, we overcome this lack of stability by presenting two multi-window approaches that rely on a "good coverage" of the time-frequency plane by the ambiguity functions of the windows. The first is to use the fractional Fourier transform of a dilated Gauss function with various angles as window functions. The essential support of a superposition of the ambiguity function from such window functions is of a "daffodil shape", which converges to a large disc as more angles are used, yielding a much broader coverage in the time-frequency domain. The second approach uses Hermite functions of various degrees as the window functions. The larger the degree, the wider the ambiguity function but with zeros on circles in the time-frequency domain. Combining Hermite functions of different degrees, we can achieve a wide coverage with zeros compensated by the essential support of the ambiguity function from other Hermite windows. Taking advantage of these multi-window procedures, we can stably perform STFT phase retrieval using the direct inversion formula. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.05486v1-abstract-full').style.display = 'none'; document.getElementById('2410.05486v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">20 pages, 11 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 42-08; 42C99; 45Q05; 49K40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.04344">arXiv:2410.04344</a> <span> [<a href="https://arxiv.org/pdf/2410.04344">pdf</a>, <a href="https://arxiv.org/format/2410.04344">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> DeepONet for Solving PDEs: Generalization Analysis in Sobolev Training </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yahong 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="2410.04344v1-abstract-short" style="display: inline;"> In this paper, we investigate the application of operator learning, specifically DeepONet, to solve partial differential equations (PDEs). Unlike function learning methods that require training separate neural networks for each PDE, operator learning generalizes across different PDEs without retraining. We focus on the performance of DeepONet in Sobolev training, addressing two key questions: the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.04344v1-abstract-full').style.display = 'inline'; document.getElementById('2410.04344v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.04344v1-abstract-full" style="display: none;"> In this paper, we investigate the application of operator learning, specifically DeepONet, to solve partial differential equations (PDEs). Unlike function learning methods that require training separate neural networks for each PDE, operator learning generalizes across different PDEs without retraining. We focus on the performance of DeepONet in Sobolev training, addressing two key questions: the approximation ability of deep branch and trunk networks, and the generalization error in Sobolev norms. Our findings highlight that deep branch networks offer significant performance benefits, while trunk networks are best kept simple. Moreover, standard sampling methods without adding derivative information in the encoding part are sufficient for minimizing generalization error in Sobolev training, based on generalization analysis. This paper fills a theoretical gap by providing error estimations for a wide range of physics-informed machine learning models and applications. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.04344v1-abstract-full').style.display = 'none'; document.getElementById('2410.04344v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.03372">arXiv:2410.03372</a> <span> [<a href="https://arxiv.org/pdf/2410.03372">pdf</a>, <a href="https://arxiv.org/format/2410.03372">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Chaotic Dynamics">nlin.CD</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Instrumentation and Methods for Astrophysics">astro-ph.IM</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"> HALO: A High-Precision Orbit Propagation Tool for Mission Design in the Cis-Lunar Domain </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Granier%2C+Q">Quentin Granier</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yang Yang</a>, <a href="/search/math?searchtype=author&query=Dempster%2C+A">Andrew Dempster</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.03372v1-abstract-short" style="display: inline;"> With the recent implementation of the Artemis Accords, interest in the cis-lunar space is rapidly increasing, necessitating the development of more precise and accurate modeling tools. While general-purpose mission design tools are available, this study proposes an open-source mission design tool, HALO, for High-precision Analyser for Lunar Orbits. This work presents a comprehensive review of the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.03372v1-abstract-full').style.display = 'inline'; document.getElementById('2410.03372v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.03372v1-abstract-full" style="display: none;"> With the recent implementation of the Artemis Accords, interest in the cis-lunar space is rapidly increasing, necessitating the development of more precise and accurate modeling tools. While general-purpose mission design tools are available, this study proposes an open-source mission design tool, HALO, for High-precision Analyser for Lunar Orbits. This work presents a comprehensive review of the modeling approaches, structural design, and algorithms employed, aiming at facilitating ease of use and adaptation for other research in the cis-lunar domain. Furthermore, accuracy studies of the propagator are provided for various orbits of interest within this domain, including low lunar orbits, elliptical frozen orbits, and 3-body problem orbits, such as Near Rectilinear Halo Orbits and Distant Retrograde Orbits. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.03372v1-abstract-full').style.display = 'none'; document.getElementById('2410.03372v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.02025">arXiv:2410.02025</a> <span> [<a href="https://arxiv.org/pdf/2410.02025">pdf</a>, <a href="https://arxiv.org/format/2410.02025">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="Artificial Intelligence">cs.AI</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"> A Likelihood Based Approach to Distribution Regression Using Conditional Deep Generative Models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Kumar%2C+S">Shivam Kumar</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yun Yang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+L">Lizhen Lin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.02025v1-abstract-short" style="display: inline;"> In this work, we explore the theoretical properties of conditional deep generative models under the statistical framework of distribution regression where the response variable lies in a high-dimensional ambient space but concentrates around a potentially lower-dimensional manifold. More specifically, we study the large-sample properties of a likelihood-based approach for estimating these models.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.02025v1-abstract-full').style.display = 'inline'; document.getElementById('2410.02025v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.02025v1-abstract-full" style="display: none;"> In this work, we explore the theoretical properties of conditional deep generative models under the statistical framework of distribution regression where the response variable lies in a high-dimensional ambient space but concentrates around a potentially lower-dimensional manifold. More specifically, we study the large-sample properties of a likelihood-based approach for estimating these models. Our results lead to the convergence rate of a sieve maximum likelihood estimator (MLE) for estimating the conditional distribution (and its devolved counterpart) of the response given predictors in the Hellinger (Wasserstein) metric. Our rates depend solely on the intrinsic dimension and smoothness of the true conditional distribution. These findings provide an explanation of why conditional deep generative models can circumvent the curse of dimensionality from the perspective of statistical foundations and demonstrate that they can learn a broader class of nearly singular conditional distributions. Our analysis also emphasizes the importance of introducing a small noise perturbation to the data when they are supported sufficiently close to a manifold. Finally, in our numerical studies, we demonstrate the effective implementation of the proposed approach using both synthetic and real-world datasets, which also provide complementary validation to our theoretical findings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.02025v1-abstract-full').style.display = 'none'; document.getElementById('2410.02025v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: text overlap with arXiv:1708.06633 by other authors</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.00229">arXiv:2410.00229</a> <span> [<a href="https://arxiv.org/pdf/2410.00229">pdf</a>, <a href="https://arxiv.org/format/2410.00229">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="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"> Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Q">Qin Li</a>, <a href="/search/math?searchtype=author&query=Oprea%2C+M">Maria Oprea</a>, <a href="/search/math?searchtype=author&query=Wang%2C+L">Li Wang</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="2410.00229v1-abstract-short" style="display: inline;"> Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with measurements. Consequently, these problems are naturally framed as stochastic inverse problems. In this paper, we explore three aspects of this problem: direct inversio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.00229v1-abstract-full').style.display = 'inline'; document.getElementById('2410.00229v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.00229v1-abstract-full" style="display: none;"> Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with measurements. Consequently, these problems are naturally framed as stochastic inverse problems. In this paper, we explore three aspects of this problem: direct inversion, variational formulation with regularization, and optimization via gradient flows, drawing parallels with deterministic inverse problems. A key difference from the deterministic case is the space in which we operate. Here, we work within probability space rather than Euclidean or Sobolev spaces, making tools from measure transport theory necessary for the study. Our findings reveal that the choice of metric -- both in the design of the loss function and in the optimization process -- significantly impacts the stability and properties of the optimizer. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.00229v1-abstract-full').style.display = 'none'; document.getElementById('2410.00229v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.20124">arXiv:2409.20124</a> <span> [<a href="https://arxiv.org/pdf/2409.20124">pdf</a>, <a href="https://arxiv.org/ps/2409.20124">ps</a>, <a href="https://arxiv.org/format/2409.20124">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"> Conditional Diffusion Models are Minimax-Optimal and Manifold-Adaptive for Conditional Distribution Estimation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Tang%2C+R">Rong Tang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+L">Lizhen Lin</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yun 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="2409.20124v1-abstract-short" style="display: inline;"> We consider a class of conditional forward-backward diffusion models for conditional generative modeling, that is, generating new data given a covariate (or control variable). To formally study the theoretical properties of these conditional generative models, we adopt a statistical framework of distribution regression to characterize the large sample properties of the conditional distribution est… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.20124v1-abstract-full').style.display = 'inline'; document.getElementById('2409.20124v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.20124v1-abstract-full" style="display: none;"> We consider a class of conditional forward-backward diffusion models for conditional generative modeling, that is, generating new data given a covariate (or control variable). To formally study the theoretical properties of these conditional generative models, we adopt a statistical framework of distribution regression to characterize the large sample properties of the conditional distribution estimators induced by these conditional forward-backward diffusion models. Here, the conditional distribution of data is assumed to smoothly change over the covariate. In particular, our derived convergence rate is minimax-optimal under the total variation metric within the regimes covered by the existing literature. Additionally, we extend our theory by allowing both the data and the covariate variable to potentially admit a low-dimensional manifold structure. In this scenario, we demonstrate that the conditional forward-backward diffusion model can adapt to both manifold structures, meaning that the derived estimation error bound (under the Wasserstein metric) depends only on the intrinsic dimensionalities of the data and the covariate. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.20124v1-abstract-full').style.display = 'none'; document.getElementById('2409.20124v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.17090">arXiv:2409.17090</a> <span> [<a href="https://arxiv.org/pdf/2409.17090">pdf</a>, <a href="https://arxiv.org/format/2409.17090">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Locally Regularized Sparse Graph by Fast Proximal Gradient Descent </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Sun%2C+D">Dongfang Sun</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yingzhen 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="2409.17090v1-abstract-short" style="display: inline;"> Sparse graphs built by sparse representation has been demonstrated to be effective in clustering high-dimensional data. Albeit the compelling empirical performance, the vanilla sparse graph ignores the geometric information of the data by performing sparse representation for each datum separately. In order to obtain a sparse graph aligned with the local geometric structure of data, we propose a no… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.17090v1-abstract-full').style.display = 'inline'; document.getElementById('2409.17090v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.17090v1-abstract-full" style="display: none;"> Sparse graphs built by sparse representation has been demonstrated to be effective in clustering high-dimensional data. Albeit the compelling empirical performance, the vanilla sparse graph ignores the geometric information of the data by performing sparse representation for each datum separately. In order to obtain a sparse graph aligned with the local geometric structure of data, we propose a novel Support Regularized Sparse Graph, abbreviated as SRSG, for data clustering. SRSG encourages local smoothness on the neighborhoods of nearby data points by a well-defined support regularization term. We propose a fast proximal gradient descent method to solve the non-convex optimization problem of SRSG with the convergence matching the Nesterov's optimal convergence rate of first-order methods on smooth and convex objective function with Lipschitz continuous gradient. Extensive experimental results on various real data sets demonstrate the superiority of SRSG over other competing clustering methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.17090v1-abstract-full').style.display = 'none'; document.getElementById('2409.17090v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Accepted by UAI2023</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.16519">arXiv:2409.16519</a> <span> [<a href="https://arxiv.org/pdf/2409.16519">pdf</a>, <a href="https://arxiv.org/format/2409.16519">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</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"> Feynman-Kac Formula for Nonlinear Schr枚dinger Equations with Applications in Numerical Approximations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cheung%2C+H">Hang Cheung</a>, <a href="/search/math?searchtype=author&query=Qiu%2C+J">Jinniao Qiu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yang 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="2409.16519v3-abstract-short" style="display: inline;"> This paper is devoted to a Feynman-Kac formula for general nonlinear time-dependent Schr枚dinger equations with applications in numerical approximations. Our formulation integrates both the Fisk-Stratonovich and It么 integrals within the framework of backward stochastic differential equations. Utilizing this Feynman-Kac representation, we propose a deep-learning-based approach for numerical approxim… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16519v3-abstract-full').style.display = 'inline'; document.getElementById('2409.16519v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.16519v3-abstract-full" style="display: none;"> This paper is devoted to a Feynman-Kac formula for general nonlinear time-dependent Schr枚dinger equations with applications in numerical approximations. Our formulation integrates both the Fisk-Stratonovich and It么 integrals within the framework of backward stochastic differential equations. Utilizing this Feynman-Kac representation, we propose a deep-learning-based approach for numerical approximation. Numerical experiments are performed to validate the accuracy and efficiency of our method, and a convergence analysis is provided to support the results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16519v3-abstract-full').style.display = 'none'; document.getElementById('2409.16519v3-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages, 4 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81Q05; 60H30; 35Q41 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.14551">arXiv:2409.14551</a> <span> [<a href="https://arxiv.org/pdf/2409.14551">pdf</a>, <a href="https://arxiv.org/format/2409.14551">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"> Unconditional energy stable IEQ-FEMs for the Cahn-Hilliard-Navier-Stokes equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yaoyao Chen</a>, <a href="/search/math?searchtype=author&query=Li%2C+D">Dongqian Li</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yin Yang</a>, <a href="/search/math?searchtype=author&query=Yin%2C+P">Peimeng Yin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.14551v1-abstract-short" style="display: inline;"> We propose several unconditionally energy stable invariant energy quadratization (IEQ) finite element methods (FEMs) to solve the Cahn-Hilliard-Navier-Stokes (CHNS) equations. The time discretization of these IEQ-FEMs is based on the first- and second-order backward differentiation methods. The intermediate function introduced by the IEQ approach is positioned in different function spaces: the con… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.14551v1-abstract-full').style.display = 'inline'; document.getElementById('2409.14551v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.14551v1-abstract-full" style="display: none;"> We propose several unconditionally energy stable invariant energy quadratization (IEQ) finite element methods (FEMs) to solve the Cahn-Hilliard-Navier-Stokes (CHNS) equations. The time discretization of these IEQ-FEMs is based on the first- and second-order backward differentiation methods. The intermediate function introduced by the IEQ approach is positioned in different function spaces: the continuous function space, and a combination of the continuous function and finite element spaces. These methods offer distinct advantages. Consequently, we propose a new hybrid IEQ-FEM that combines the strengths of both schemes, offering computational efficiency and unconditional energy stability in the finite element space. We provide rigorous proofs of mass conservation and energy dissipation for the proposed IEQ-FEMs. Several numerical experiments are presented to validate the accuracy, efficiency, and solution properties of the proposed IEQ-FEMs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.14551v1-abstract-full').style.display = 'none'; document.getElementById('2409.14551v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages, 13 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/2409.13544">arXiv:2409.13544</a> <span> [<a href="https://arxiv.org/pdf/2409.13544">pdf</a>, <a href="https://arxiv.org/format/2409.13544">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Graph Similarity Regularized Softmax for Semi-Supervised Node Classification </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yiming Yang</a>, <a href="/search/math?searchtype=author&query=Liu%2C+J">Jun Liu</a>, <a href="/search/math?searchtype=author&query=Wan%2C+W">Wei Wan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.13544v1-abstract-short" style="display: inline;"> Graph Neural Networks (GNNs) are powerful deep learning models designed for graph-structured data, demonstrating effectiveness across a wide range of applications.The softmax function is the most commonly used classifier for semi-supervised node classification. However, the softmax function lacks spatial information of the graph structure. In this paper, we propose a graph similarity regularized s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.13544v1-abstract-full').style.display = 'inline'; document.getElementById('2409.13544v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.13544v1-abstract-full" style="display: none;"> Graph Neural Networks (GNNs) are powerful deep learning models designed for graph-structured data, demonstrating effectiveness across a wide range of applications.The softmax function is the most commonly used classifier for semi-supervised node classification. However, the softmax function lacks spatial information of the graph structure. In this paper, we propose a graph similarity regularized softmax for GNNs in semi-supervised node classification. By incorporating non-local total variation (TV) regularization into the softmax activation function, we can more effectively capture the spatial information inherent in graphs. The weights in the non-local gradient and divergence operators are determined based on the graph's adjacency matrix. We apply the proposed method into the architecture of GCN and GraphSAGE, testing them on citation and webpage linking datasets, respectively. Numerical experiments demonstrate its good performance in node classification and generalization capabilities. These results indicate that the graph similarity regularized softmax is effective on both assortative and disassortative graphs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.13544v1-abstract-full').style.display = 'none'; document.getElementById('2409.13544v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.09104">arXiv:2409.09104</a> <span> [<a href="https://arxiv.org/pdf/2409.09104">pdf</a>, <a href="https://arxiv.org/ps/2409.09104">ps</a>, <a href="https://arxiv.org/format/2409.09104">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"> Hybrid LSMR algorithms for large-scale general-form regularization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yanfei 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="2409.09104v1-abstract-short" style="display: inline;"> The hybrid LSMR algorithm is proposed for large-scale general-form regularization. It is based on a Krylov subspace projection method where the matrix $A$ is first projected onto a subspace, typically a Krylov subspace, which is implemented via the Golub-Kahan bidiagonalization process applied to $A$, with starting vector $b$. Then a regularization term is employed to the projections. Finally, an… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.09104v1-abstract-full').style.display = 'inline'; document.getElementById('2409.09104v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.09104v1-abstract-full" style="display: none;"> The hybrid LSMR algorithm is proposed for large-scale general-form regularization. It is based on a Krylov subspace projection method where the matrix $A$ is first projected onto a subspace, typically a Krylov subspace, which is implemented via the Golub-Kahan bidiagonalization process applied to $A$, with starting vector $b$. Then a regularization term is employed to the projections. Finally, an iterative algorithm is exploited to solve a least squares problem with constraints. The resulting algorithms are called the {hybrid LSMR algorithm}. At every step, we exploit LSQR algorithm to solve the inner least squares problem, which is proven to become better conditioned as the number of $k$ increases, so that the LSQR algorithm converges faster. We prove how to select the stopping tolerances for LSQR in order to guarantee that the regularized solution obtained by iteratively computing the inner least squares problems and the one obtained by exactly computing the inner least squares problems have the same accuracy. Numerical experiments illustrate that the best regularized solution by the hybrid LSMR algorithm is as accurate as that by JBDQR which is a joint bidiagonalization based algorithm. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.09104v1-abstract-full').style.display = 'none'; document.getElementById('2409.09104v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65F22; 65F10; 65J20; 65F35; 65F50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.08768">arXiv:2409.08768</a> <span> [<a href="https://arxiv.org/pdf/2409.08768">pdf</a>, <a href="https://arxiv.org/format/2409.08768">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</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="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Measure-Theoretic Time-Delay Embedding </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Botvinick-Greenhouse%2C+J">Jonah Botvinick-Greenhouse</a>, <a href="/search/math?searchtype=author&query=Oprea%2C+M">Maria Oprea</a>, <a href="/search/math?searchtype=author&query=Maulik%2C+R">Romit Maulik</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="2409.08768v1-abstract-short" style="display: inline;"> The celebrated Takens' embedding theorem provides a theoretical foundation for reconstructing the full state of a dynamical system from partial observations. However, the classical theorem assumes that the underlying system is deterministic and that observations are noise-free, limiting its applicability in real-world scenarios. Motivated by these limitations, we rigorously establish a measure-the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.08768v1-abstract-full').style.display = 'inline'; document.getElementById('2409.08768v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.08768v1-abstract-full" style="display: none;"> The celebrated Takens' embedding theorem provides a theoretical foundation for reconstructing the full state of a dynamical system from partial observations. However, the classical theorem assumes that the underlying system is deterministic and that observations are noise-free, limiting its applicability in real-world scenarios. Motivated by these limitations, we rigorously establish a measure-theoretic generalization that adopts an Eulerian description of the dynamics and recasts the embedding as a pushforward map between probability spaces. Our mathematical results leverage recent advances in optimal transportation theory. Building on our novel measure-theoretic time-delay embedding theory, we have developed a new computational framework that forecasts the full state of a dynamical system from time-lagged partial observations, engineered with better robustness to handle sparse and noisy data. We showcase the efficacy and versatility of our approach through several numerical examples, ranging from the classic Lorenz-63 system to large-scale, real-world applications such as NOAA sea surface temperature forecasting and ERA5 wind field reconstruction. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.08768v1-abstract-full').style.display = 'none'; document.getElementById('2409.08768v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">32 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/2409.07152">arXiv:2409.07152</a> <span> [<a href="https://arxiv.org/pdf/2409.07152">pdf</a>, <a href="https://arxiv.org/ps/2409.07152">ps</a>, <a href="https://arxiv.org/format/2409.07152">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 bounds for a class of singular integral operators in variable exponent Herz-Morrey spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yanqi Yang</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Q">Qi 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="2409.07152v1-abstract-short" style="display: inline;"> Let T be the singular integral operator with variable kernel defined by $Tf(x)= p.v. \int_{\mathbb{R}^{n}}K(x,x-y)f(y)\mathrm{d}y$ and $D^纬(0\leq纬\leq1)$ be the fractional differentiation operator, where $K(x,z)=\frac{惟(x,z')}{|z|^{n}}$, $z'=\frac{z}{|z|},~~z\neq0$. Let $~T^{\ast}~$and $~T^\sharp~$ be the adjoint of $T$ and the pseudo-adjoint of $T$, respectively. In this paper, via the expansion… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.07152v1-abstract-full').style.display = 'inline'; document.getElementById('2409.07152v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.07152v1-abstract-full" style="display: none;"> Let T be the singular integral operator with variable kernel defined by $Tf(x)= p.v. \int_{\mathbb{R}^{n}}K(x,x-y)f(y)\mathrm{d}y$ and $D^纬(0\leq纬\leq1)$ be the fractional differentiation operator, where $K(x,z)=\frac{惟(x,z')}{|z|^{n}}$, $z'=\frac{z}{|z|},~~z\neq0$. Let $~T^{\ast}~$and $~T^\sharp~$ be the adjoint of $T$ and the pseudo-adjoint of $T$, respectively. In this paper, via the expansion of spherical harmonics and the estimates of the convolution operators $T_{m,j}$, we shall prove some boundedness results for $TD^纬-D^纬T$ and $(T^{\ast}-T^{\sharp})D^纬$ under natural regularity assumptions on the exponent function on a class of generalized Herz-Morrey spaces with weight and variable exponent, which extend some known results. Moreover, various norm characterizations for the product $T_{1}T_{2}$ and the pseudo-product $T_{1}\circ T_{2}$ are also established. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.07152v1-abstract-full').style.display = 'none'; document.getElementById('2409.07152v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.04663">arXiv:2409.04663</a> <span> [<a href="https://arxiv.org/pdf/2409.04663">pdf</a>, <a href="https://arxiv.org/format/2409.04663">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"> On pattern formation in the thermodynamically-consistent variational Gray-Scott model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hao%2C+W">Wenrui Hao</a>, <a href="/search/math?searchtype=author&query=Liu%2C+C">Chun Liu</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yiwei Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yahong 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="2409.04663v1-abstract-short" style="display: inline;"> In this paper, we explore pattern formation in a four-species variational Gary-Scott model, which includes all reverse reactions and introduces a virtual species to describe the birth-death process in the classical Gray-Scott model. This modification transforms the classical Gray-Scott model into a thermodynamically consistent closed system. The classical two-species Gray-Scott model can be viewed… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04663v1-abstract-full').style.display = 'inline'; document.getElementById('2409.04663v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.04663v1-abstract-full" style="display: none;"> In this paper, we explore pattern formation in a four-species variational Gary-Scott model, which includes all reverse reactions and introduces a virtual species to describe the birth-death process in the classical Gray-Scott model. This modification transforms the classical Gray-Scott model into a thermodynamically consistent closed system. The classical two-species Gray-Scott model can be viewed as a subsystem of the variational Gray-Scott model in the limiting case when the small parameter $蔚$, related to the reaction rate of the reverse reactions, approaches zero. We numerically study the physically more complete Gray-Scott model with various $蔚$ in one dimension. By decreasing $蔚$, we observed that the stationary pattern in the classical Gray-Scott model can be stabilized as the transient state in the variational model for a significantly small $蔚$. Additionally, the variational model admits oscillated and traveling-wave-like pattern for small $蔚$. The persistent time of these patterns is on the order of $O(蔚^{-1})$. We also analyze the stability of two uniform steady states in the variational Gary-Scott model for fixed $蔚$. Although both states are stable in a certain sense, the gradient flow type dynamics of the variational model exhibit a selection effect based on the initial conditions, with pattern formation occurring only if the initial condition does not converge to the boundary steady state, which corresponds to the trivial uniform steady state in the classical Gray-Scott model. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04663v1-abstract-full').style.display = 'none'; document.getElementById('2409.04663v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, 11 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/2409.02585">arXiv:2409.02585</a> <span> [<a href="https://arxiv.org/pdf/2409.02585">pdf</a>, <a href="https://arxiv.org/ps/2409.02585">ps</a>, <a href="https://arxiv.org/format/2409.02585">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Four fault-free $B_{n-2}$'s in $B_{n}$ under the random node fault model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Meng%2C+K">KaiYue Meng</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuxing 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="2409.02585v1-abstract-short" style="display: inline;"> Let $n\geq 4$. Each $B_{n-2}$ in $B_n$ has one of the forms $a_1a_2X^{n-2}$, $a_1X^{n-2}a_2$ and $X^{n-2}a_1a_2$. Let $1-p$ be the fault probiability of each node in the $n$-dimensional bubble-sort network $B_{n}$ under the random node fault model. In this paper, we determine the probability that there are four distinct fault-free $B_{n-2}$'s in $B_{n}$ by considering all possible combinatorial ca… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.02585v1-abstract-full').style.display = 'inline'; document.getElementById('2409.02585v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.02585v1-abstract-full" style="display: none;"> Let $n\geq 4$. Each $B_{n-2}$ in $B_n$ has one of the forms $a_1a_2X^{n-2}$, $a_1X^{n-2}a_2$ and $X^{n-2}a_1a_2$. Let $1-p$ be the fault probiability of each node in the $n$-dimensional bubble-sort network $B_{n}$ under the random node fault model. In this paper, we determine the probability that there are four distinct fault-free $B_{n-2}$'s in $B_{n}$ by considering all possible combinatorial cases of the four fault-free $B_{n-2}$'s. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.02585v1-abstract-full').style.display = 'none'; document.getElementById('2409.02585v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">63 pages, no figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C90 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.00901">arXiv:2409.00901</a> <span> [<a href="https://arxiv.org/pdf/2409.00901">pdf</a>, <a href="https://arxiv.org/format/2409.00901">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yunfei 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="2409.00901v2-abstract-short" style="display: inline;"> This paper studies the problem of how efficiently functions in the Sobolev spaces $\mathcal{W}^{s,q}([0,1]^d)$ and Besov spaces $\mathcal{B}^s_{q,r}([0,1]^d)$ can be approximated by deep ReLU neural networks with width $W$ and depth $L$, when the error is measured in the $L^p([0,1]^d)$ norm. This problem has been studied by several recent works, which obtained the approximation rate… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.00901v2-abstract-full').style.display = 'inline'; document.getElementById('2409.00901v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.00901v2-abstract-full" style="display: none;"> This paper studies the problem of how efficiently functions in the Sobolev spaces $\mathcal{W}^{s,q}([0,1]^d)$ and Besov spaces $\mathcal{B}^s_{q,r}([0,1]^d)$ can be approximated by deep ReLU neural networks with width $W$ and depth $L$, when the error is measured in the $L^p([0,1]^d)$ norm. This problem has been studied by several recent works, which obtained the approximation rate $\mathcal{O}((WL)^{-2s/d})$ up to logarithmic factors when $p=q=\infty$, and the rate $\mathcal{O}(L^{-2s/d})$ for networks with fixed width when the Sobolev embedding condition $1/q -1/p<s/d$ holds. We generalize these results by showing that the rate $\mathcal{O}((WL)^{-2s/d})$ indeed holds under the Sobolev embedding condition. It is known that this rate is optimal up to logarithmic factors. The key tool in our proof is a novel encoding of sparse vectors by using deep ReLU neural networks with varied width and depth, which may be of independent interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.00901v2-abstract-full').style.display = 'none'; document.getElementById('2409.00901v2-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.00692">arXiv:2409.00692</a> <span> [<a href="https://arxiv.org/pdf/2409.00692">pdf</a>, <a href="https://arxiv.org/ps/2409.00692">ps</a>, <a href="https://arxiv.org/format/2409.00692">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Every nonsymmetric $4$-class association scheme can be generated by a digraph </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuefeng 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="2409.00692v1-abstract-short" style="display: inline;"> A (di)graph $螕$ generates a commutative association scheme $\mathfrak{X}$ if and only if the adjacency matrix of $螕$ generates the Bose-Mesner algebra of $\mathfrak{X}$. In [17, Theorem 1.1], Monzillo and Penji膰 proved that, except for amorphic symmetric association schemes, every $3$-class association scheme can be generated by an adjacency matrix of a (di)graph. In this paper, we characterize wh… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.00692v1-abstract-full').style.display = 'inline'; document.getElementById('2409.00692v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.00692v1-abstract-full" style="display: none;"> A (di)graph $螕$ generates a commutative association scheme $\mathfrak{X}$ if and only if the adjacency matrix of $螕$ generates the Bose-Mesner algebra of $\mathfrak{X}$. In [17, Theorem 1.1], Monzillo and Penji膰 proved that, except for amorphic symmetric association schemes, every $3$-class association scheme can be generated by an adjacency matrix of a (di)graph. In this paper, we characterize when a commutative association scheme with exactly one pair of nonsymmetric relations can be generated by a digraph under the certain assumptions. As an application, we show that each nonsymmetric $4$-class association scheme can be generated by a digraph. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.00692v1-abstract-full').style.display = 'none'; document.getElementById('2409.00692v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.12759">arXiv:2408.12759</a> <span> [<a href="https://arxiv.org/pdf/2408.12759">pdf</a>, <a href="https://arxiv.org/format/2408.12759">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> An Inverse Hyperbolic Problem with Application to Joint Photoacoustic Parameter Determination </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+S">Shitao Liu</a>, <a href="/search/math?searchtype=author&query=Uhlmann%2C+G">Gunther Uhlmann</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yang 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="2408.12759v1-abstract-short" style="display: inline;"> We consider an inverse problem of recovering a parameter appearing in all levels in a second-order hyperbolic equation from a single boundary measurement. The model is motivated from applications in photoacoustic tomography when one seeks to recover both the wave speed and the initial ultrasound pressure from a single ultrasound signal. In particular, our result shows that the ratio of the initial… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.12759v1-abstract-full').style.display = 'inline'; document.getElementById('2408.12759v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.12759v1-abstract-full" style="display: none;"> We consider an inverse problem of recovering a parameter appearing in all levels in a second-order hyperbolic equation from a single boundary measurement. The model is motivated from applications in photoacoustic tomography when one seeks to recover both the wave speed and the initial ultrasound pressure from a single ultrasound signal. In particular, our result shows that the ratio of the initial ultrasound pressure and the wave speed squared uniquely determines both of them respectively. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.12759v1-abstract-full').style.display = 'none'; document.getElementById('2408.12759v1-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">19 pages. arXiv admin note: text overlap with arXiv:2210.03865</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R30; 35L05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.09435">arXiv:2408.09435</a> <span> [<a href="https://arxiv.org/pdf/2408.09435">pdf</a>, <a href="https://arxiv.org/format/2408.09435">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> A modified Ricci flow on arbitrary weighted graph </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jicheng Ma</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yunyan 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="2408.09435v1-abstract-short" style="display: inline;"> In this paper, we propose a modified Ricci flow, as well as a quasi-normalized Ricci flow, on arbitrary weighted graph. Each of these two flows has a unique global solution. In particular, these global existence and uniqueness results do not require an exit condition proposed by Bai et al in a recent work [2]. As applications, these two Ricci flows are applied to community detection for complex ne… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.09435v1-abstract-full').style.display = 'inline'; document.getElementById('2408.09435v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.09435v1-abstract-full" style="display: none;"> In this paper, we propose a modified Ricci flow, as well as a quasi-normalized Ricci flow, on arbitrary weighted graph. Each of these two flows has a unique global solution. In particular, these global existence and uniqueness results do not require an exit condition proposed by Bai et al in a recent work [2]. As applications, these two Ricci flows are applied to community detection for complex networks, including Karate Club, American football games, Facebook, as well as artificial networks. In our algorithms, unlike in [5,15], there is no need to perform surgery at every iteration, only one surgery needs to be performed after the last iteration. From three commonly used criteria for evaluating community detection algorithms, ARI, NMI and Q, we conclude that our algorithms outperform existing algorithms, including Ollivier's Ricci flow [5], normalized Ollivier's Ricci flow and normalized Lin-Lu-Yau's Ricci flow [15]. The codes for our algorithms are available at https://github.com/mjc191812/Modified-Ricci-Flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.09435v1-abstract-full').style.display = 'none'; document.getElementById('2408.09435v1-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, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">15 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C21; 05C85; 35R02; 68Q06 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.02931">arXiv:2408.02931</a> <span> [<a href="https://arxiv.org/pdf/2408.02931">pdf</a>, <a href="https://arxiv.org/ps/2408.02931">ps</a>, <a href="https://arxiv.org/format/2408.02931">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Weakly distance-regular digraphs whose underlying graphs are distance-regular,II </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zeng%2C+Q">Qing Zeng</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuefeng Yang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+K">Kaishun Wang</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="2408.02931v1-abstract-short" style="display: inline;"> Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In [16], we classified all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded n-cubes, or Doob graphs. In this paper, we classify all commutative weakly distance-regular digraphs whose underlying graphs are Johnson graphs or folded Johnson graphs. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.02931v1-abstract-full" style="display: none;"> Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In [16], we classified all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded n-cubes, or Doob graphs. In this paper, we classify all commutative weakly distance-regular digraphs whose underlying graphs are Johnson graphs or folded Johnson graphs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02931v1-abstract-full').style.display = 'none'; document.getElementById('2408.02931v1-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 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/2408.02902">arXiv:2408.02902</a> <span> [<a href="https://arxiv.org/pdf/2408.02902">pdf</a>, <a href="https://arxiv.org/ps/2408.02902">ps</a>, <a href="https://arxiv.org/format/2408.02902">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Fractional Laplace operator and related Schr枚dinger equations on locally finite graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+M">Mengjie Zhang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+Y">Yong Lin</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yunyan 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="2408.02902v1-abstract-short" style="display: inline;"> In this paper, we first define a discrete version of the fractional Laplace operator $(-螖)^{s}$ through the heat semigroup on a stochastically complete, connected, locally finite graph $G = (V, E, 渭, w)$. Secondly, we define the fractional divergence and give another form of $(-螖)^s$. The third point, and the foremost, is the introduction of the fractional Sobolev space $W^{s,2}(V)$, which is nece… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02902v1-abstract-full').style.display = 'inline'; document.getElementById('2408.02902v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.02902v1-abstract-full" style="display: none;"> In this paper, we first define a discrete version of the fractional Laplace operator $(-螖)^{s}$ through the heat semigroup on a stochastically complete, connected, locally finite graph $G = (V, E, 渭, w)$. Secondly, we define the fractional divergence and give another form of $(-螖)^s$. The third point, and the foremost, is the introduction of the fractional Sobolev space $W^{s,2}(V)$, which is necessary when we study problems involving $(-螖)^{s}$. Finally, using the mountain-pass theorem and the Nehari manifold, we obtain multiplicity solutions to a discrete fractional Schr枚dinger equation on $G$. We caution the readers that though these existence results are well known in the continuous case, the discrete case is quite different. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02902v1-abstract-full').style.display = 'none'; document.getElementById('2408.02902v1-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">23 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/2408.02864">arXiv:2408.02864</a> <span> [<a href="https://arxiv.org/pdf/2408.02864">pdf</a>, <a href="https://arxiv.org/ps/2408.02864">ps</a>, <a href="https://arxiv.org/format/2408.02864">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</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="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Distributions in spaces with thick submanifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ding%2C+J">Jiajia Ding</a>, <a href="/search/math?searchtype=author&query=Vindas%2C+J">Jasson Vindas</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yunyun 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="2408.02864v1-abstract-short" style="display: inline;"> This article generalizes the results of [J. Math. Anal. Appl. 512 (2022), Article No. 126075], which presented a theory of distributions (generalized functions) with a singular curve contained in the domain of the test functions. In this present article we construct a theory of distributions in $\mathbb{R}^n$ with a ``thick submanifold'', that is, a new theory of thick distributions in… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02864v1-abstract-full').style.display = 'inline'; document.getElementById('2408.02864v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.02864v1-abstract-full" style="display: none;"> This article generalizes the results of [J. Math. Anal. Appl. 512 (2022), Article No. 126075], which presented a theory of distributions (generalized functions) with a singular curve contained in the domain of the test functions. In this present article we construct a theory of distributions in $\mathbb{R}^n$ with a ``thick submanifold'', that is, a new theory of thick distributions in $\mathbb{R}^n$ whose domain contains a submanifold on which test functions may be singular. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02864v1-abstract-full').style.display = 'none'; document.getElementById('2408.02864v1-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46F05; 46F10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.00543">arXiv:2408.00543</a> <span> [<a href="https://arxiv.org/pdf/2408.00543">pdf</a>, <a href="https://arxiv.org/format/2408.00543">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Global convergence of a modified BFGS-type method based on function information for nonconvex multiobjective optimization problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yingxue 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="2408.00543v2-abstract-short" style="display: inline;"> In this paper, based on function information, we propose a modified BFGS-type method for nonconvex multiobjective optimization problems (MFQNMO). In the multiobjective quasi-Newton method (QNMO), each iteration involves separately approximating the Hessian matrix for each component objective function, which results in significant storage and computational burdens. MFQNMO employs a common BFGS-type… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.00543v2-abstract-full').style.display = 'inline'; document.getElementById('2408.00543v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.00543v2-abstract-full" style="display: none;"> In this paper, based on function information, we propose a modified BFGS-type method for nonconvex multiobjective optimization problems (MFQNMO). In the multiobjective quasi-Newton method (QNMO), each iteration involves separately approximating the Hessian matrix for each component objective function, which results in significant storage and computational burdens. MFQNMO employs a common BFGS-type matrix to approximate the Hessian matrix of all objective functions in each iteration. This matrix is updated using function information from the previous step. This approach strikes a balance between efficacy and computational cost. We confirm the convergence of the method without relying on convexity assumptions, under mild conditions, we establish a local superlinear convergence rate for MFQNMO. Furthermore, we validate its effectiveness through experiments on both nonconvex and convex test problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.00543v2-abstract-full').style.display = 'none'; document.getElementById('2408.00543v2-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.15384">arXiv:2407.15384</a> <span> [<a href="https://arxiv.org/pdf/2407.15384">pdf</a>, <a href="https://arxiv.org/format/2407.15384">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Inversion Diameter and Treewidth </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yichen Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+H">Haozhe Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuxuan Yang</a>, <a href="/search/math?searchtype=author&query=Lu%2C+M">Mei Lu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.15384v1-abstract-short" style="display: inline;"> In an oriented graph $\overrightarrow{G}$, the inversion of a subset $X$ of vertices consists in reversing the orientation of all arcs with both end-vertices in $X$. The inversion graph of a graph $G$, denoted by $\mathcal{I}(G)$, is the graph whose vertices are orientations of $G$ in which two orientations $\overrightarrow{G_1}$ and $\overrightarrow{G_2}$ are adjacent if and only if there is an i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.15384v1-abstract-full').style.display = 'inline'; document.getElementById('2407.15384v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.15384v1-abstract-full" style="display: none;"> In an oriented graph $\overrightarrow{G}$, the inversion of a subset $X$ of vertices consists in reversing the orientation of all arcs with both end-vertices in $X$. The inversion graph of a graph $G$, denoted by $\mathcal{I}(G)$, is the graph whose vertices are orientations of $G$ in which two orientations $\overrightarrow{G_1}$ and $\overrightarrow{G_2}$ are adjacent if and only if there is an inversion $X$ transforming $\overrightarrow{G_1}$ into $\overrightarrow{G_2}$. The inversion diameter of a graph $G$ is the diameter of its inversion graph $\mathcal{I}(G)$ denoted by $diam(\mathcal{I}(G))$. Havet, H枚rsch, and Rambaud~(2024) first proved that for $G$ of treewidth $k$, $diam(\mathcal{I}(G)) \le 2k$, and there are graphs of treewidth $k$ with inversion diameter $k+2$. In this paper, we construct graphs of treewidth $k$ with inversion diameter $2k$, which implies that the previous upper bound $diam(\mathcal{I}(G)) \le 2k$ is tight. Moreover, for graphs with maximum degree $螖$, Havet, H枚rsch, and Rambaud~(2024) proved $diam(\mathcal{I}(G)) \le 2螖-1$ and conjectured that $diam(\mathcal{I}(G)) \le 螖$. We prove the conjecture when $螖=3$ with the help of computer calculations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.15384v1-abstract-full').style.display = 'none'; document.getElementById('2407.15384v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.12527">arXiv:2407.12527</a> <span> [<a href="https://arxiv.org/pdf/2407.12527">pdf</a>, <a href="https://arxiv.org/format/2407.12527">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</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="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computational Physics">physics.comp-ph</span> </div> </div> <p class="title is-5 mathjax"> Random ordinate method for mitigating the ray effect in radiative transport equation simulations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+L">Lei Li</a>, <a href="/search/math?searchtype=author&query=Tang%2C+M">Min Tang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuqi 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="2407.12527v1-abstract-short" style="display: inline;"> The Discrete Ordinates Method (DOM) is the most widely used velocity discretization method for simulating the radiative transport equation. The ray effect stands as a long-standing drawback of DOM. In benchmark tests displaying the ray effect, we observe low regularity in velocity within the solution. To address this issue, we propose a random ordinate method (ROM) to mitigate the ray effect. Comp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.12527v1-abstract-full').style.display = 'inline'; document.getElementById('2407.12527v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.12527v1-abstract-full" style="display: none;"> The Discrete Ordinates Method (DOM) is the most widely used velocity discretization method for simulating the radiative transport equation. The ray effect stands as a long-standing drawback of DOM. In benchmark tests displaying the ray effect, we observe low regularity in velocity within the solution. To address this issue, we propose a random ordinate method (ROM) to mitigate the ray effect. Compared with other strategies proposed in the literature for mitigating the ray effect, ROM offers several advantages: 1) the computational cost is comparable to DOM; 2) it is simple and requires minimal changes to existing code based on DOM; 3) it is easily parallelizable and independent of the problem setup. Analytical results are presented for the convergence orders of the error and bias, and numerical tests demonstrate its effectiveness in mitigating the ray effect. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.12527v1-abstract-full').style.display = 'none'; document.getElementById('2407.12527v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.11353">arXiv:2407.11353</a> <span> [<a href="https://arxiv.org/pdf/2407.11353">pdf</a>, <a href="https://arxiv.org/format/2407.11353">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"> Preconditioned Gradient Descent Finds Over-Parameterized Neural Networks with Sharp Generalization for Nonparametric Regression </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yingzhen 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="2407.11353v1-abstract-short" style="display: inline;"> We consider nonparametric regression by an over-parameterized two-layer neural network trained by gradient descent (GD) or its variant in this paper. We show that, if the neural network is trained with a novel Preconditioned Gradient Descent (PGD) with early stopping and the target function has spectral bias widely studied in the deep learning literature, the trained network renders a particularly… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.11353v1-abstract-full').style.display = 'inline'; document.getElementById('2407.11353v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.11353v1-abstract-full" style="display: none;"> We consider nonparametric regression by an over-parameterized two-layer neural network trained by gradient descent (GD) or its variant in this paper. We show that, if the neural network is trained with a novel Preconditioned Gradient Descent (PGD) with early stopping and the target function has spectral bias widely studied in the deep learning literature, the trained network renders a particularly sharp generalization bound with a minimax optimal rate of $\cO({1}/{n^{4伪/(4伪+1)}})$, which is sharper the current standard rate of $\cO({1}/{n^{2伪/(2伪+1)}})$ with $2伪= d/(d-1)$ when the data is distributed uniformly on the unit sphere in $\RR^d$ and $n$ is the size of the training data. When the target function has no spectral bias, we prove that neural network trained with regular GD with early stopping still enjoys minimax optimal rate, and in this case our results do not require distributional assumptions in contrast with the current known results. Our results are built upon two significant technical contributions. First, uniform convergence to the NTK is established during the training process by PGD or GD, so that we can have a nice decomposition of the neural network function at any step of GD or PGD into a function in the RKHS and an error function with a small $L^{\infty}$-norm. Second, local Rademacher complexity is employed to tightly bound the Rademacher complexity of the function class comprising all the possible neural network functions obtained by GD or PGD. Our results also indicate that PGD can be another way of avoiding the usual linear regime of NTK and obtaining sharper generalization bound, because PGD induces a different kernel with lower kernel complexity during the training than the regular NTK induced by the network architecture trained by regular GD. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.11353v1-abstract-full').style.display = 'none'; document.getElementById('2407.11353v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.09764">arXiv:2407.09764</a> <span> [<a href="https://arxiv.org/pdf/2407.09764">pdf</a>, <a href="https://arxiv.org/ps/2407.09764">ps</a>, <a href="https://arxiv.org/format/2407.09764">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Restricted Cohomology of Heisenberg Lie Algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yong 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="2407.09764v1-abstract-short" style="display: inline;"> The Heisenberg Lie algebras over an algebraically closed field F of characteristic p > 0 always admit a family of restricted Lie algebras. We use the ordinary 1- and 2-cohomology spaces with adjoint coefficients to compute the restricted 1- and 2-cohomology spaces of these restricted Heisenberg Lie algebras. We describe the infinitesimal restricted deformations of the restricted Lie algebras. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.09764v1-abstract-full" style="display: none;"> The Heisenberg Lie algebras over an algebraically closed field F of characteristic p > 0 always admit a family of restricted Lie algebras. We use the ordinary 1- and 2-cohomology spaces with adjoint coefficients to compute the restricted 1- and 2-cohomology spaces of these restricted Heisenberg Lie algebras. We describe the infinitesimal restricted deformations of the restricted Lie algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.09764v1-abstract-full').style.display = 'none'; document.getElementById('2407.09764v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: text overlap with arXiv:2402.14249. arXiv admin note: text overlap with arXiv:2402.14249</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B50; 17B56 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.08237">arXiv:2407.08237</a> <span> [<a href="https://arxiv.org/pdf/2407.08237">pdf</a>, <a href="https://arxiv.org/ps/2407.08237">ps</a>, <a href="https://arxiv.org/format/2407.08237">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Associated Mersenne graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+J">Jianxin Wei</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yujun 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="2407.08237v1-abstract-short" style="display: inline;"> In this paper, a new sub-family of Hypercubes called the \textit{associated Mersenne graphs} $\mathcal{M}_{n}$ are introduced. The definition of associated Mersenne graphs is motivated from the Fibonacci-run graphs ({脰}. E千ecio千lu, V. Ir拧i膷, 2021) by extending run-constrained strings to circularly-run-constrained strings. The name of this new family of graphs is identified with the interesting fac… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.08237v1-abstract-full').style.display = 'inline'; document.getElementById('2407.08237v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.08237v1-abstract-full" style="display: none;"> In this paper, a new sub-family of Hypercubes called the \textit{associated Mersenne graphs} $\mathcal{M}_{n}$ are introduced. The definition of associated Mersenne graphs is motivated from the Fibonacci-run graphs ({脰}. E千ecio千lu, V. Ir拧i膷, 2021) by extending run-constrained strings to circularly-run-constrained strings. The name of this new family of graphs is identified with the interesting fact that $|V(\mathcal{M}_{n})|$ is equal to the $n$-th associated Mersenne number. Various interesting structural and enumerative properties of associated Mersenne graphs are investigated, including the analogue of the fundamental recursion, number of vertices and edges, radius, diameter, center, periphery and medianicity. Some future research directions and open problems concerning associated Mersenne graphs are also proposed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.08237v1-abstract-full').style.display = 'none'; document.getElementById('2407.08237v1-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, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.05077">arXiv:2407.05077</a> <span> [<a href="https://arxiv.org/pdf/2407.05077">pdf</a>, <a href="https://arxiv.org/ps/2407.05077">ps</a>, <a href="https://arxiv.org/format/2407.05077">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> </div> </div> <p class="title is-5 mathjax"> Regularity of powers of edge ideals of edge-weighted integrally closed cycles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhu%2C+G">Guangjun Zhu</a>, <a href="/search/math?searchtype=author&query=Cui%2C+Y">Yijun Cui</a>, <a href="/search/math?searchtype=author&query=Li%2C+J">Jiaxin Li</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yi 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="2407.05077v1-abstract-short" style="display: inline;"> This paper gives exact formulas for the regularity of powers of edge ideals of an edge-weighted integrally closed cycle. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.05077v1-abstract-full" style="display: none;"> This paper gives exact formulas for the regularity of powers of edge ideals of an edge-weighted integrally closed cycle. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.05077v1-abstract-full').style.display = 'none'; document.getElementById('2407.05077v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: text overlap with arXiv:2403.03609, arXiv:2401.02111</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 13B22; 13F20; Secondary 05C99; 05E40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.00288">arXiv:2407.00288</a> <span> [<a href="https://arxiv.org/pdf/2407.00288">pdf</a>, <a href="https://arxiv.org/ps/2407.00288">ps</a>, <a href="https://arxiv.org/format/2407.00288">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> A Rank-Two Case of Local-Global Compatibility for $l = p$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuji 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="2407.00288v2-abstract-short" style="display: inline;"> We prove the classical $l = p$ local-global compatibility conjecture for certain regular algebraic cuspidal automorphic representations of weight 0 for GL$_2$ over CM fields. Using an automorphy lifting theorem, we show that if the automorphic side comes from a twist of Steinberg at $v | l$, then the Galois side has nontrivial monodromy at $v$. Based on this observation, we will give a definition… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.00288v2-abstract-full').style.display = 'inline'; document.getElementById('2407.00288v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.00288v2-abstract-full" style="display: none;"> We prove the classical $l = p$ local-global compatibility conjecture for certain regular algebraic cuspidal automorphic representations of weight 0 for GL$_2$ over CM fields. Using an automorphy lifting theorem, we show that if the automorphic side comes from a twist of Steinberg at $v | l$, then the Galois side has nontrivial monodromy at $v$. Based on this observation, we will give a definition of the Fontaine-Mazur $\mathcal{L}$-invariants attached to certain automorphic representations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.00288v2-abstract-full').style.display = 'none'; document.getElementById('2407.00288v2-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 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">9 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/2406.18231">arXiv:2406.18231</a> <span> [<a href="https://arxiv.org/pdf/2406.18231">pdf</a>, <a href="https://arxiv.org/ps/2406.18231">ps</a>, <a href="https://arxiv.org/format/2406.18231">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Return time sets and product recurrence </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+J">Jian Li</a>, <a href="/search/math?searchtype=author&query=Liang%2C+X">Xianjuan Liang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yini 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="2406.18231v1-abstract-short" style="display: inline;"> Let $G$ be a countable infinite discrete group. We show that a subset $F$ of $G$ contains a return time set of some piecewise syndetic recurrent point $x$ in a compact Hausdorff space $X$ with a $G$-action if and only if $F$ is a quasi-central set. As an application, we show that if a nonempty closed subsemigroup $S$ of the Stone-膶ech compactification $尾G$ contains the smallest ideal $K(尾G)$ of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.18231v1-abstract-full').style.display = 'inline'; document.getElementById('2406.18231v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.18231v1-abstract-full" style="display: none;"> Let $G$ be a countable infinite discrete group. We show that a subset $F$ of $G$ contains a return time set of some piecewise syndetic recurrent point $x$ in a compact Hausdorff space $X$ with a $G$-action if and only if $F$ is a quasi-central set. As an application, we show that if a nonempty closed subsemigroup $S$ of the Stone-膶ech compactification $尾G$ contains the smallest ideal $K(尾G)$ of $尾G$ then $S$-product recurrent is equivalent to distality, which partially answers a question of Auslander and Furstenberg (Trans. Amer. Math. Soc. 343, 1994, 221--232). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.18231v1-abstract-full').style.display = 'none'; document.getElementById('2406.18231v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.13989">arXiv:2406.13989</a> <span> [<a href="https://arxiv.org/pdf/2406.13989">pdf</a>, <a href="https://arxiv.org/format/2406.13989">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="Information Theory">cs.IT</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"> Random pairing MLE for estimation of item parameters in Rasch model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuepeng Yang</a>, <a href="/search/math?searchtype=author&query=Ma%2C+C">Cong Ma</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="2406.13989v1-abstract-short" style="display: inline;"> The Rasch model, a classical model in the item response theory, is widely used in psychometrics to model the relationship between individuals' latent traits and their binary responses on assessments or questionnaires. In this paper, we introduce a new likelihood-based estimator -- random pairing maximum likelihood estimator ($\mathsf{RP\text{-}MLE}$) and its bootstrapped variant multiple random pa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.13989v1-abstract-full').style.display = 'inline'; document.getElementById('2406.13989v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.13989v1-abstract-full" style="display: none;"> The Rasch model, a classical model in the item response theory, is widely used in psychometrics to model the relationship between individuals' latent traits and their binary responses on assessments or questionnaires. In this paper, we introduce a new likelihood-based estimator -- random pairing maximum likelihood estimator ($\mathsf{RP\text{-}MLE}$) and its bootstrapped variant multiple random pairing MLE ($\mathsf{MRP\text{-}MLE}$) that faithfully estimate the item parameters in the Rasch model. The new estimators have several appealing features compared to existing ones. First, both work for sparse observations, an increasingly important scenario in the big data era. Second, both estimators are provably minimax optimal in terms of finite sample $\ell_{\infty}$ estimation error. Lastly, $\mathsf{RP\text{-}MLE}$ admits precise distributional characterization that allows uncertainty quantification on the item parameters, e.g., construction of confidence intervals of the item parameters. The main idea underlying $\mathsf{RP\text{-}MLE}$ and $\mathsf{MRP\text{-}MLE}$ is to randomly pair user-item responses to form item-item comparisons. This is carefully designed to reduce the problem size while retaining statistical independence. We also provide empirical evidence of the efficacy of the two new estimators using both simulated and real data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.13989v1-abstract-full').style.display = 'none'; document.getElementById('2406.13989v1-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.11152">arXiv:2406.11152</a> <span> [<a href="https://arxiv.org/pdf/2406.11152">pdf</a>, <a href="https://arxiv.org/format/2406.11152">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"> Limit Results for Estimation of Connectivity Matrix in Multi-layer Stochastic Block Models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Su%2C+W">Wenqing Su</a>, <a href="/search/math?searchtype=author&query=Guo%2C+X">Xiao Guo</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Ying 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="2406.11152v1-abstract-short" style="display: inline;"> Multi-layer networks arise naturally in various domains including biology, finance and sociology, among others. The multi-layer stochastic block model (multi-layer SBM) is commonly used for community detection in the multi-layer networks. Most of current literature focuses on statistical consistency of community detection methods under multi-layer SBMs. However, the asymptotic distributional prope… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.11152v1-abstract-full').style.display = 'inline'; document.getElementById('2406.11152v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.11152v1-abstract-full" style="display: none;"> Multi-layer networks arise naturally in various domains including biology, finance and sociology, among others. The multi-layer stochastic block model (multi-layer SBM) is commonly used for community detection in the multi-layer networks. Most of current literature focuses on statistical consistency of community detection methods under multi-layer SBMs. However, the asymptotic distributional properties are also indispensable which play an important role in statistical inference. In this work, we aim to study the estimation and asymptotic properties of the layer-wise scaled connectivity matrices in the multi-layer SBMs. We develop a novel and efficient method to estimate the scaled connectivity matrices. Under the multi-layer SBM and its variant multi-layer degree-corrected SBM, we establish the asymptotic normality of the estimated matrices under mild conditions, which can be used for interval estimation and hypothesis testing. Simulations show the superior performance of proposed method over existing methods in two considered statistical inference tasks. We also apply the method to a real dataset and obtain interpretable results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.11152v1-abstract-full').style.display = 'none'; document.getElementById('2406.11152v1-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Yang%2C+Y&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Yang%2C+Y&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a 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