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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=Wei%2C+Y&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Wei%2C+Y&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Wei%2C+Y&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Wei%2C+Y&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> <li> <a href="/search/?searchtype=author&query=Wei%2C+Y&start=150" class="pagination-link " aria-label="Page 4" aria-current="page">4 </a> </li> <li> <a href="/search/?searchtype=author&query=Wei%2C+Y&start=200" class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.16196">arXiv:2503.16196</a> <span> [<a href="https://arxiv.org/pdf/2503.16196">pdf</a>, <a href="https://arxiv.org/ps/2503.16196">ps</a>, <a href="https://arxiv.org/format/2503.16196">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"> An interior penalty DG method with correct and minimal averages, jumps and penalties for the miscible displacement problem of nonnegative characteristic form, and SUPG-type error estimates under low regularity, dominating Darcy velocity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Du%2C+Z">Zhijie Du</a>, <a href="/search/math?searchtype=author&query=Duan%2C+H">Huoyuan Duan</a>, <a href="/search/math?searchtype=author&query=Tan%2C+R+C+E">Roger C E Tan</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuanhong Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.16196v1-abstract-short" style="display: inline;"> An interior penalty DG method is proposed for the steady-state linear partial differential equations of nonnegative characteristic form, suitable for mixed second-order elliptic-parabolic and first-order hyperbolic equations. Due to the different natures of the elliptic, parabolic, and hyperbolic equations. In the new DG method, the averages, jumps and penalties are minimal, correctly and only imp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.16196v1-abstract-full').style.display = 'inline'; document.getElementById('2503.16196v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.16196v1-abstract-full" style="display: none;"> An interior penalty DG method is proposed for the steady-state linear partial differential equations of nonnegative characteristic form, suitable for mixed second-order elliptic-parabolic and first-order hyperbolic equations. Due to the different natures of the elliptic, parabolic, and hyperbolic equations. In the new DG method, the averages, jumps and penalties are minimal, correctly and only imposed on the diffusion-diffusion element boundaries, in addition to the well-known upwind jumps associating with the advection velocity. For the advection-dominated problem, the penalties can be further reduced only being imposed on the diffusion-dominated subset of the diffusion-diffusion element boundaries.This is based on the novel, crucial technique about the multiple partitions of the set of the interelement boundaries into a number of subsets with respect to the diffusion and to the advection and on the consistency result we have proven. The new DG method is the first DG method and the first time that the continuity and discontinuity of the solution are correctly identified and justified of the general steady-state linear partial differential equations of nonnegative characteristic form. The new DG method and its analysis are applied to the miscible displacement problem of vanishing diffusion coefficient and of low regularity, dominating Darcy flow velocity which lives in $H(\operatorname{div};惟)\cap \prod_{j=1}^J (H^r(D_j))^d$ for $r<1$ other than the usual assumption $(W^{1,\infty}(惟))^d$. We prove the SUPG-type error estimates $\mathcal{O}(h^{\ell+\frac{1}{2}})$ for any element polynomial of degree $\ell\ge 1$ on generally shaped and nonconforming meshes, where the convergence order is independent of the regularity of the advection velocity. The SUPG-type error estimates obtained are new and the first time known under the low regularity of the advection velocity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.16196v1-abstract-full').style.display = 'none'; document.getElementById('2503.16196v1-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 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.15884">arXiv:2503.15884</a> <span> [<a href="https://arxiv.org/pdf/2503.15884">pdf</a>, <a href="https://arxiv.org/ps/2503.15884">ps</a>, <a href="https://arxiv.org/format/2503.15884">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Upper bounds for the Alexandrov-Fenchel deficit via integral formulas </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Kwong%2C+K">Kwok-Kun Kwong</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yong Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.15884v1-abstract-short" style="display: inline;"> We derive a number of sharp upper bounds for the deficit in the Alexandrov-Fenchel inequality using a weighted Minkowski integral formula and an integral formula for the deficit in Jensen's inequality. Our estimates yield results under weaker convexity assumptions compared to approaches based on inverse curvature flows. The use of weighted formulas provides flexibility in deriving inequalities wit… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.15884v1-abstract-full').style.display = 'inline'; document.getElementById('2503.15884v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.15884v1-abstract-full" style="display: none;"> We derive a number of sharp upper bounds for the deficit in the Alexandrov-Fenchel inequality using a weighted Minkowski integral formula and an integral formula for the deficit in Jensen's inequality. Our estimates yield results under weaker convexity assumptions compared to approaches based on inverse curvature flows. The use of weighted formulas provides flexibility in deriving inequalities with different weight functions. Furthermore, our estimates are more quantitative as they include a distance term measuring the domain's deviation from a reference ball. We also analyze the stability of a weighted geometric inequality from a recent paper \cite{kwong2023geometric} via analysis of the support function on the sphere and show that, with an optimal choice of the origin, this inequality is stronger than the classical isoperimetric inequality. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.15884v1-abstract-full').style.display = 'none'; document.getElementById('2503.15884v1-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 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C42; 53C24 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.15188">arXiv:2503.15188</a> <span> [<a href="https://arxiv.org/pdf/2503.15188">pdf</a>, <a href="https://arxiv.org/ps/2503.15188">ps</a>, <a href="https://arxiv.org/format/2503.15188">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"> Convergence analysis of SPH method on irregular particle distributions for the Poisson equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Qiao%2C+Z">Zhonghua Qiao</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yifan Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.15188v1-abstract-short" style="display: inline;"> The accuracy of particle approximation in Smoothed Particle Hydrodynamics (SPH) method decreases due to irregular particle distributions, especially for second-order derivatives. This study aims to enhance the accuracy of SPH method and analyze its convergence with irregular particle distributions. By establishing regularity conditions for particle distributions, we ensure that the local truncatio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.15188v1-abstract-full').style.display = 'inline'; document.getElementById('2503.15188v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.15188v1-abstract-full" style="display: none;"> The accuracy of particle approximation in Smoothed Particle Hydrodynamics (SPH) method decreases due to irregular particle distributions, especially for second-order derivatives. This study aims to enhance the accuracy of SPH method and analyze its convergence with irregular particle distributions. By establishing regularity conditions for particle distributions, we ensure that the local truncation error of traditional SPH formulations, including first and second derivatives, achieves second-order accuracy. Our proposed method, the volume reconstruction SPH method, guarantees these regularity conditions while preserving the discrete maximum principle. Benefiting from the discrete maximum principle, we conduct a rigorous global error analysis in the $L^\infty$-norm for the Poisson equation with variable coefficients, achieving second-order convergence. Numerical examples are presented to validate the theoretical findings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.15188v1-abstract-full').style.display = 'none'; document.getElementById('2503.15188v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35J05; 65M12; 65M15; 65M75; 76M28 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.07066">arXiv:2502.07066</a> <span> [<a href="https://arxiv.org/pdf/2502.07066">pdf</a>, <a href="https://arxiv.org/format/2502.07066">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Cryptography and Security">cs.CR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> General-Purpose $f$-DP Estimation and Auditing in a Black-Box Setting </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Askin%2C+%C3%96">脰nder Askin</a>, <a href="/search/math?searchtype=author&query=Dette%2C+H">Holger Dette</a>, <a href="/search/math?searchtype=author&query=Dunsche%2C+M">Martin Dunsche</a>, <a href="/search/math?searchtype=author&query=Kutta%2C+T">Tim Kutta</a>, <a href="/search/math?searchtype=author&query=Lu%2C+Y">Yun Lu</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yu Wei</a>, <a href="/search/math?searchtype=author&query=Zikas%2C+V">Vassilis Zikas</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="2502.07066v1-abstract-short" style="display: inline;"> In this paper we propose new methods to statistically assess $f$-Differential Privacy ($f$-DP), a recent refinement of differential privacy (DP) that remedies certain weaknesses of standard DP (including tightness under algorithmic composition). A challenge when deploying differentially private mechanisms is that DP is hard to validate, especially in the black-box setting. This has led to numerous… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.07066v1-abstract-full').style.display = 'inline'; document.getElementById('2502.07066v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.07066v1-abstract-full" style="display: none;"> In this paper we propose new methods to statistically assess $f$-Differential Privacy ($f$-DP), a recent refinement of differential privacy (DP) that remedies certain weaknesses of standard DP (including tightness under algorithmic composition). A challenge when deploying differentially private mechanisms is that DP is hard to validate, especially in the black-box setting. This has led to numerous empirical methods for auditing standard DP, while $f$-DP remains less explored. We introduce new black-box methods for $f$-DP that, unlike existing approaches for this privacy notion, do not require prior knowledge of the investigated algorithm. Our procedure yields a complete estimate of the $f$-DP trade-off curve, with theoretical guarantees of convergence. Additionally, we propose an efficient auditing method that empirically detects $f$-DP violations with statistical certainty, merging techniques from non-parametric estimation and optimal classification theory. Through experiments on a range of DP mechanisms, we demonstrate the effectiveness of our estimation and auditing procedures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.07066v1-abstract-full').style.display = 'none'; document.getElementById('2502.07066v1-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 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages, 32 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/2501.07904">arXiv:2501.07904</a> <span> [<a href="https://arxiv.org/pdf/2501.07904">pdf</a>, <a href="https://arxiv.org/format/2501.07904">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"> Effective algorithms for tensor train decomposition via the UTV framework </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yuchao Wang</a>, <a href="/search/math?searchtype=author&query=Che%2C+M">Maolin Che</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.07904v1-abstract-short" style="display: inline;"> The tensor train (TT) decomposition is used to compress large tensors into a more compact form by exploiting their inherent data structures. A fundamental approach for constructing the TT format is the TT-SVD, which extracts the TT-cores by the singular value decompositions (SVDs) sequentially. But in practical applications, it is often not necessary to compute full SVDs. In this article, we there… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.07904v1-abstract-full').style.display = 'inline'; document.getElementById('2501.07904v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.07904v1-abstract-full" style="display: none;"> The tensor train (TT) decomposition is used to compress large tensors into a more compact form by exploiting their inherent data structures. A fundamental approach for constructing the TT format is the TT-SVD, which extracts the TT-cores by the singular value decompositions (SVDs) sequentially. But in practical applications, it is often not necessary to compute full SVDs. In this article, we therefore propose a new method called the TT-UTV. It utilizes the virtues of rank-revealing UTV decomposition to compute the TT format for a large-scale tensor, hence requires less computational cost. We analyze the error bounds on the accuracy of these algorithms both in the URV and ULV cases, then recommend different sweep patterns for these two cases. We perform numerical experiments on some applications, including magnetic resonance imaging (MRI) data completion, to illustrate their good performance in practice. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.07904v1-abstract-full').style.display = 'none'; document.getElementById('2501.07904v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 15A69; 15A72; 65F10; 65F15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.04324">arXiv:2501.04324</a> <span> [<a href="https://arxiv.org/pdf/2501.04324">pdf</a>, <a href="https://arxiv.org/ps/2501.04324">ps</a>, <a href="https://arxiv.org/format/2501.04324">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"> LU Decomposition and Generalized Autoone-Takagi Decomposition of Dual Matrices and their Applications </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xu%2C+R">Renjie Xu</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</a>, <a href="/search/math?searchtype=author&query=Yan%2C+H">Hong Yan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.04324v1-abstract-short" style="display: inline;"> This paper uses matrix transformations to provide the Autoone-Takagi decomposition of dual complex symmetric matrices and extends it to dual quaternion $畏$-Hermitian matrices. The LU decomposition of dual matrices is given using the general solution of the Sylvester equation, and its equivalence to the existence of rank-k decomposition and dual Moore-Penrose generalized inverse (DMPGI) is proved.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.04324v1-abstract-full').style.display = 'inline'; document.getElementById('2501.04324v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.04324v1-abstract-full" style="display: none;"> This paper uses matrix transformations to provide the Autoone-Takagi decomposition of dual complex symmetric matrices and extends it to dual quaternion $畏$-Hermitian matrices. The LU decomposition of dual matrices is given using the general solution of the Sylvester equation, and its equivalence to the existence of rank-k decomposition and dual Moore-Penrose generalized inverse (DMPGI) is proved. Similar methods are then used to provide the Cholesky decomposition of dual real symmetric positive definite matrices. Both of our decompositions are driven by applications in numerical linear algebra. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.04324v1-abstract-full').style.display = 'none'; document.getElementById('2501.04324v1-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 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages, 0 figures. This paper has been accepted for publication in Linear and Multilinear Algebra. The final published version may differ</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 15A23; 15B33; 65F55 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.01002">arXiv:2501.01002</a> <span> [<a href="https://arxiv.org/pdf/2501.01002">pdf</a>, <a href="https://arxiv.org/format/2501.01002">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"> Multi-Objective Optimization-Based Anonymization of Structured Data for Machine Learning </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yusi Wei</a>, <a href="/search/math?searchtype=author&query=Benson%2C+H+Y">Hande Y. Benson</a>, <a href="/search/math?searchtype=author&query=Agor%2C+J+K">Joseph K. Agor</a>, <a href="/search/math?searchtype=author&query=Capan%2C+M">Muge Capan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.01002v1-abstract-short" style="display: inline;"> Data is essential for secondary use, but ensuring its privacy while allowing such use is a critical challenge. Various techniques have been proposed to address privacy concerns in data sharing and publishing. However, these methods often degrade data utility, impacting the performance of machine learning (ML) models. Our research identifies key limitations in existing optimization models for priva… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.01002v1-abstract-full').style.display = 'inline'; document.getElementById('2501.01002v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.01002v1-abstract-full" style="display: none;"> Data is essential for secondary use, but ensuring its privacy while allowing such use is a critical challenge. Various techniques have been proposed to address privacy concerns in data sharing and publishing. However, these methods often degrade data utility, impacting the performance of machine learning (ML) models. Our research identifies key limitations in existing optimization models for privacy preservation, particularly in handling categorical variables, assessing data utility, and evaluating effectiveness across diverse datasets. We propose a novel multi-objective optimization model that simultaneously minimizes information loss and maximizes protection against attacks. This model is empirically validated using diverse datasets and compared with two existing algorithms. We assess information loss, the number of individuals subject to linkage or homogeneity attacks, and ML performance after anonymization. The results indicate that our model achieves lower information loss and more effectively mitigates the risk of attacks, reducing the number of individuals susceptible to these attacks compared to alternative algorithms in some cases. Additionally, our model maintains comparative ML performance relative to the original data or data anonymized by other methods. Our findings highlight significant improvements in privacy protection and ML model performance, offering a comprehensive framework for balancing privacy and utility in data sharing. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.01002v1-abstract-full').style.display = 'none'; document.getElementById('2501.01002v1-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 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.08923">arXiv:2412.08923</a> <span> [<a href="https://arxiv.org/pdf/2412.08923">pdf</a>, <a href="https://arxiv.org/ps/2412.08923">ps</a>, <a href="https://arxiv.org/format/2412.08923">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Alexandrov-Fenchel type inequalities with convex weight in space forms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Kwong%2C+K">Kwok-Kun Kwong</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yong Wei</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="2412.08923v1-abstract-short" style="display: inline;"> In this paper, we derive new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces under various convexity assumptions in Euclidean, spherical, and hyperbolic spaces. These inequalities extend classical results by incorporating weights given by convex, non-decreasing positive functions, which are otherwise arbitrary. Our approach gives rise to a broad family… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.08923v1-abstract-full').style.display = 'inline'; document.getElementById('2412.08923v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.08923v1-abstract-full" style="display: none;"> In this paper, we derive new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces under various convexity assumptions in Euclidean, spherical, and hyperbolic spaces. These inequalities extend classical results by incorporating weights given by convex, non-decreasing positive functions, which are otherwise arbitrary. Our approach gives rise to a broad family of geometric inequalities, as each convex, non-decreasing function yields a corresponding inequality, providing considerable flexibility. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.08923v1-abstract-full').style.display = 'none'; document.getElementById('2412.08923v1-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 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</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.08039">arXiv:2412.08039</a> <span> [<a href="https://arxiv.org/pdf/2412.08039">pdf</a>, <a href="https://arxiv.org/ps/2412.08039">ps</a>, <a href="https://arxiv.org/format/2412.08039">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 priori estimates and moving plane method for a class of Grushin equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bauer%2C+W">Wolfram Bauer</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yawei Wei</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+X">Xiaodong 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="2412.08039v1-abstract-short" style="display: inline;"> In this paper, we study three kinds of nonlinear degenerate elliptic equations containing the Grushin operator. First, we prove radial symmetry and a decay rate at infinity of solutions to such a Grushin equation by using the moving plane method in combination with suitable integral inequalities. Applying similar methods, we obtain nonexistence results for solutions to a second type of Grushin equ… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.08039v1-abstract-full').style.display = 'inline'; document.getElementById('2412.08039v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.08039v1-abstract-full" style="display: none;"> In this paper, we study three kinds of nonlinear degenerate elliptic equations containing the Grushin operator. First, we prove radial symmetry and a decay rate at infinity of solutions to such a Grushin equation by using the moving plane method in combination with suitable integral inequalities. Applying similar methods, we obtain nonexistence results for solutions to a second type of Grushin equation in Euclidean space and in half space, respectively. Finally, we derive a priori estimates for positive solutions to more general types of Grushin equations by employing blow up analysis. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.08039v1-abstract-full').style.display = 'none'; document.getElementById('2412.08039v1-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35J70; 35B45; 35A16 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.17328">arXiv:2411.17328</a> <span> [<a href="https://arxiv.org/pdf/2411.17328">pdf</a>, <a href="https://arxiv.org/ps/2411.17328">ps</a>, <a href="https://arxiv.org/format/2411.17328">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="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+T">Tianci Luo</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yong Wei</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="2411.17328v1-abstract-short" style="display: inline;"> The horospherical $p$-Christoffel-Minkowski problem was posed by Li and Xu (2022) as a problem prescribing the $k$-th horospherical $p$-surface area measure of $h$-convex domains in hyperbolic space $\mathbb{H}^{n+1}$. It is a natural generalization of the classical $L^p$ Christoffel-Minkowski problem in the Euclidean space $\mathbb{R}^{n+1}$. In this paper, we consider a fully nonlinear equation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.17328v1-abstract-full').style.display = 'inline'; document.getElementById('2411.17328v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.17328v1-abstract-full" style="display: none;"> The horospherical $p$-Christoffel-Minkowski problem was posed by Li and Xu (2022) as a problem prescribing the $k$-th horospherical $p$-surface area measure of $h$-convex domains in hyperbolic space $\mathbb{H}^{n+1}$. It is a natural generalization of the classical $L^p$ Christoffel-Minkowski problem in the Euclidean space $\mathbb{R}^{n+1}$. In this paper, we consider a fully nonlinear equation associated with the horospherical $p$-Christoffel-Minkowski problem. We establish the existence of a uniformly $h$-convex solution under appropriate assumptions on the prescribed function. The key to the proof is the full rank theorem, which we will demonstrate using a viscosity approach based on the idea of Bryan-Ivaki-Scheuer (2023). When $p=0$, the horospherical $p$-Christoffel-Minkowski problem in $\mathbb{H}^{n+1}$ is equivalent to a Nirenberg-type problem on $\mathbb{S}^n$ in conformal geometry. Therefore, our result implies the existence of solutions to the Nirenberg-type problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.17328v1-abstract-full').style.display = 'none'; document.getElementById('2411.17328v1-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 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">25 pages, submitted</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C42; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.09930">arXiv:2411.09930</a> <span> [<a href="https://arxiv.org/pdf/2411.09930">pdf</a>, <a href="https://arxiv.org/ps/2411.09930">ps</a>, <a href="https://arxiv.org/format/2411.09930">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"> On some regularity properties of mixed local and nonlocal elliptic equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Su%2C+X">Xifeng Su</a>, <a href="/search/math?searchtype=author&query=Valdinoci%2C+E">Enrico Valdinoci</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuanhong Wei</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jiwen Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.09930v1-abstract-short" style="display: inline;"> This article is concerned with ``up to $C^{2, 伪}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an estimate on the $L^\infty$ norm of weak solutions is established for more general cases than the ones present in the literature, including here critical nonlinearities… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.09930v1-abstract-full').style.display = 'inline'; document.getElementById('2411.09930v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.09930v1-abstract-full" style="display: none;"> This article is concerned with ``up to $C^{2, 伪}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an estimate on the $L^\infty$ norm of weak solutions is established for more general cases than the ones present in the literature, including here critical nonlinearities. We then prove the interior $C^{1,伪}$-regularity and the $C^{1,伪}$-regularity up to the boundary of weak solutions, which extends previous results by the authors [X. Su, E. Valdinoci, Y. Wei and J. Zhang, Math. Z. (2022)], where the nonlinearities considered were of subcritical type. In addition, we establish the interior $C^{2,伪}$-regularity of solutions for all $s\in(0,1)$ and the $C^{2,伪}$-regularity up to the boundary for all $s\in(0,\frac{1}{2})$, with sharp regularity exponents. For further perusal, we also include a strong maximum principle and some properties about the principal eigenvalue. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.09930v1-abstract-full').style.display = 'none'; document.getElementById('2411.09930v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 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">Journal of Differential Equations</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35B65; 35R11; 35J67 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.08377">arXiv:2411.08377</a> <span> [<a href="https://arxiv.org/pdf/2411.08377">pdf</a>, <a href="https://arxiv.org/format/2411.08377">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"> Dual-Valued Functions of Dual Matrices with Applications in Causal Emergence </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+T">Tong Wei</a>, <a href="/search/math?searchtype=author&query=Ding%2C+W">Weiyang Ding</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</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="2411.08377v1-abstract-short" style="display: inline;"> Dual continuation, an innovative insight into extending the real-valued functions of real matrices to the dual-valued functions of dual matrices with a foundation of the G芒teaux derivative, is proposed. Theoretically, the general forms of dual-valued vector and matrix norms, the remaining properties in the real field, are provided. In particular, we focus on the dual-valued vector $p$-norm… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.08377v1-abstract-full').style.display = 'inline'; document.getElementById('2411.08377v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.08377v1-abstract-full" style="display: none;"> Dual continuation, an innovative insight into extending the real-valued functions of real matrices to the dual-valued functions of dual matrices with a foundation of the G芒teaux derivative, is proposed. Theoretically, the general forms of dual-valued vector and matrix norms, the remaining properties in the real field, are provided. In particular, we focus on the dual-valued vector $p$-norm $(1\!\leq\! p\!\leq\!\infty)$ and the unitarily invariant dual-valued Ky Fan $p$-$k$-norm $(1\!\leq\! p\!\leq\!\infty)$. The equivalence between the dual-valued Ky Fan $p$-$k$-norm and the dual-valued vector $p$-norm of the first $k$ singular values of the dual matrix is then demonstrated. Practically, we define the dual transitional probability matrix (DTPM), as well as its dual-valued effective information (${\rm{EI_d}}$). Additionally, we elucidate the correlation between the ${\rm{EI_d}}$, the dual-valued Schatten $p$-norm, and the dynamical reversibility of a DTPM. Through numerical experiments on a dumbbell Markov chain, our findings indicate that the value of $k$, corresponding to the maximum value of the infinitesimal part of the dual-valued Ky Fan $p$-$k$-norm by adjusting $p$ in the interval $[1,2)$, characterizes the optimal classification number of the system for the occurrence of the causal emergence. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.08377v1-abstract-full').style.display = 'none'; document.getElementById('2411.08377v1-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">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/2411.08271">arXiv:2411.08271</a> <span> [<a href="https://arxiv.org/pdf/2411.08271">pdf</a>, <a href="https://arxiv.org/ps/2411.08271">ps</a>, <a href="https://arxiv.org/format/2411.08271">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"> High-order and Mass-conservative Regularized Implicit-explicit relaxation Runge-Kutta methods for the logarithmic Schr枚dinger equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yan%2C+J">Jingye Yan</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Hong Zhang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yabing Wei</a>, <a href="/search/math?searchtype=author&query=Qian%2C+X">Xu 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="2411.08271v1-abstract-short" style="display: inline;"> The non-differentiability of the singular nonlinearity (such as $f=\ln|u|^2$) at $u=0$ presents significant challenges in devising accurate and efficient numerical schemes for the logarithmic Schr枚dinger equation (LogSE). To address this singularity, we propose an energy regularization technique for the LogSE. For the regularized model, we utilize Implicit-Explicit Relaxation Runge-Kutta methods,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.08271v1-abstract-full').style.display = 'inline'; document.getElementById('2411.08271v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.08271v1-abstract-full" style="display: none;"> The non-differentiability of the singular nonlinearity (such as $f=\ln|u|^2$) at $u=0$ presents significant challenges in devising accurate and efficient numerical schemes for the logarithmic Schr枚dinger equation (LogSE). To address this singularity, we propose an energy regularization technique for the LogSE. For the regularized model, we utilize Implicit-Explicit Relaxation Runge-Kutta methods, which are linearly implicit, high-order, and mass-conserving for temporal discretization, in conjunction with the Fourier pseudo-spectral method in space. Ultimately, numerical results are presented to validate the efficiency of the proposed methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.08271v1-abstract-full').style.display = 'none'; document.getElementById('2411.08271v1-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 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.18784">arXiv:2410.18784</a> <span> [<a href="https://arxiv.org/pdf/2410.18784">pdf</a>, <a href="https://arxiv.org/ps/2410.18784">ps</a>, <a href="https://arxiv.org/format/2410.18784">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="Signal Processing">eess.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <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"> Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+Z">Zhihan Huang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuting Wei</a>, <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yuxin Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.18784v2-abstract-short" style="display: inline;"> The denoising diffusion probabilistic model (DDPM) has emerged as a mainstream generative model in generative AI. While sharp convergence guarantees have been established for the DDPM, the iteration complexity is, in general, proportional to the ambient data dimension, resulting in overly conservative theory that fails to explain its practical efficiency. This has motivated the recent work Li and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.18784v2-abstract-full').style.display = 'inline'; document.getElementById('2410.18784v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.18784v2-abstract-full" style="display: none;"> The denoising diffusion probabilistic model (DDPM) has emerged as a mainstream generative model in generative AI. While sharp convergence guarantees have been established for the DDPM, the iteration complexity is, in general, proportional to the ambient data dimension, resulting in overly conservative theory that fails to explain its practical efficiency. This has motivated the recent work Li and Yan (2024a) to investigate how the DDPM can achieve sampling speed-ups through automatic exploitation of intrinsic low dimensionality of data. We strengthen this line of work by demonstrating, in some sense, optimal adaptivity to unknown low dimensionality. For a broad class of data distributions with intrinsic dimension $k$, we prove that the iteration complexity of the DDPM scales nearly linearly with $k$, which is optimal when using KL divergence to measure distributional discrepancy. Notably, our work is closely aligned with the independent concurrent work Potaptchik et al. (2024) -- posted two weeks prior to ours -- in establishing nearly linear-$k$ convergence guarantees for the DDPM. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.18784v2-abstract-full').style.display = 'none'; document.getElementById('2410.18784v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 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> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.08423">arXiv:2410.08423</a> <span> [<a href="https://arxiv.org/pdf/2410.08423">pdf</a>, <a href="https://arxiv.org/format/2410.08423">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="Statistical Mechanics">cond-mat.stat-mech</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="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> </div> </div> <p class="title is-5 mathjax"> A phase transition in sampling from Restricted Boltzmann Machines </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Kwon%2C+Y">Youngwoo Kwon</a>, <a href="/search/math?searchtype=author&query=Qin%2C+Q">Qian Qin</a>, <a href="/search/math?searchtype=author&query=Wang%2C+G">Guanyang Wang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuchen Wei</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="2410.08423v1-abstract-short" style="display: inline;"> Restricted Boltzmann Machines are a class of undirected graphical models that play a key role in deep learning and unsupervised learning. In this study, we prove a phase transition phenomenon in the mixing time of the Gibbs sampler for a one-parameter Restricted Boltzmann Machine. Specifically, the mixing time varies logarithmically, polynomially, and exponentially with the number of vertices depe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.08423v1-abstract-full').style.display = 'inline'; document.getElementById('2410.08423v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.08423v1-abstract-full" style="display: none;"> Restricted Boltzmann Machines are a class of undirected graphical models that play a key role in deep learning and unsupervised learning. In this study, we prove a phase transition phenomenon in the mixing time of the Gibbs sampler for a one-parameter Restricted Boltzmann Machine. Specifically, the mixing time varies logarithmically, polynomially, and exponentially with the number of vertices depending on whether the parameter $c$ is above, equal to, or below a critical value $c_\star\approx-5.87$. A key insight from our analysis is the link between the Gibbs sampler and a dynamical system, which we utilize to quantify the former based on the behavior of the latter. To study the critical case $c= c_\star$, we develop a new isoperimetric inequality for the sampler's stationary distribution by showing that the distribution is nearly log-concave. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.08423v1-abstract-full').style.display = 'none'; document.getElementById('2410.08423v1-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">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">43 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/2410.02422">arXiv:2410.02422</a> <span> [<a href="https://arxiv.org/pdf/2410.02422">pdf</a>, <a href="https://arxiv.org/format/2410.02422">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"> Where's Ben Nevis? A 2D optimisation benchmark with 957,174 local optima based on Great Britain terrain data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuhang Wei</a>, <a href="/search/math?searchtype=author&query=Clerx%2C+M">Michael Clerx</a>, <a href="/search/math?searchtype=author&query=Mirams%2C+G+R">Gary R. Mirams</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="2410.02422v1-abstract-short" style="display: inline;"> We present a novel optimisation benchmark based on the real landscape of Great Britain (GB). The elevation data from the UK Ordnance Survey Terrain 50 dataset is slightly modified and linearly interpolated to produce a target function that simulates the GB terrain, packaged in a new Python module nevis. We introduce a discrete approach to classifying local optima and their corresponding basins of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.02422v1-abstract-full').style.display = 'inline'; document.getElementById('2410.02422v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.02422v1-abstract-full" style="display: none;"> We present a novel optimisation benchmark based on the real landscape of Great Britain (GB). The elevation data from the UK Ordnance Survey Terrain 50 dataset is slightly modified and linearly interpolated to produce a target function that simulates the GB terrain, packaged in a new Python module nevis. We introduce a discrete approach to classifying local optima and their corresponding basins of attraction, identifying 957,174 local optima of the target function. We then develop a benchmarking framework for optimisation methods based on this target function, where we propose a Generalised Expected Running Time performance measure to enable meaningful comparisons even when algorithms do not achieve successful runs (find Ben Nevis). Hyperparameter tuning is managed using the optuna framework, and plots and animations are produced to visualise algorithm performance. Using the proposed framework, we benchmark six optimisation algorithms implemented by common Python modules. Amongst those tested, the Differential Evolution algorithm implemented by scipy is the most effective for navigating the complex GB landscape and finding the summit of Ben Nevis. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.02422v1-abstract-full').style.display = 'none'; document.getElementById('2410.02422v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 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">MSC Class:</span> 90C26; 65K05 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> G.1.6 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.15693">arXiv:2409.15693</a> <span> [<a href="https://arxiv.org/pdf/2409.15693">pdf</a>, <a href="https://arxiv.org/format/2409.15693">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic">math.LO</span> </div> </div> <p class="title is-5 mathjax"> Synthetic Homotopy Theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuhang Wei</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="2409.15693v1-abstract-short" style="display: inline;"> The goal of this dissertation is to present results from synthetic homotopy theory based on homotopy type theory (HoTT). After an introduction to Martin-L枚f's dependent type theory and homotopy type theory, key results include a synthetic construction of the Hopf fibration, a proof of the Blakers--Massey theorem, and a derivation of the Freudenthal suspension theorem, with calculations of some hom… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.15693v1-abstract-full').style.display = 'inline'; document.getElementById('2409.15693v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.15693v1-abstract-full" style="display: none;"> The goal of this dissertation is to present results from synthetic homotopy theory based on homotopy type theory (HoTT). After an introduction to Martin-L枚f's dependent type theory and homotopy type theory, key results include a synthetic construction of the Hopf fibration, a proof of the Blakers--Massey theorem, and a derivation of the Freudenthal suspension theorem, with calculations of some homotopy groups of $n$-spheres. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.15693v1-abstract-full').style.display = 'none'; document.getElementById('2409.15693v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 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">Master's dissertation, 51 pages, 11 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 55U35; 55P99; 03B38 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.11039">arXiv:2409.11039</a> <span> [<a href="https://arxiv.org/pdf/2409.11039">pdf</a>, <a href="https://arxiv.org/ps/2409.11039">ps</a>, <a href="https://arxiv.org/format/2409.11039">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"> Existence of multiple radial solutions for nonlinear equation involving the mean curvature operator in Lorentz-Minkowski space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zelati%2C+V+C">Vittorio Coti Zelati</a>, <a href="/search/math?searchtype=author&query=Dong%2C+X">Xu Dong</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuanhong Wei</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="2409.11039v1-abstract-short" style="display: inline;"> We prove existence of multiple radial solutions to the Dirichlet problem for nonlinear equations involving the mean curvature operator in Lorentz-Minkowski space and a nonlinear term of concave-convex type. Solutions are found using Szulkin's critical point theory for non-smooth functional. Multiplicity results are also given for some cases in which the nonlinearity depends also on the gradient of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.11039v1-abstract-full').style.display = 'inline'; document.getElementById('2409.11039v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.11039v1-abstract-full" style="display: none;"> We prove existence of multiple radial solutions to the Dirichlet problem for nonlinear equations involving the mean curvature operator in Lorentz-Minkowski space and a nonlinear term of concave-convex type. Solutions are found using Szulkin's critical point theory for non-smooth functional. Multiplicity results are also given for some cases in which the nonlinearity depends also on the gradient of the solution. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.11039v1-abstract-full').style.display = 'none'; document.getElementById('2409.11039v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">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> 35J62 35B09 35J20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.07007">arXiv:2409.07007</a> <span> [<a href="https://arxiv.org/pdf/2409.07007">pdf</a>, <a href="https://arxiv.org/ps/2409.07007">ps</a>, <a href="https://arxiv.org/format/2409.07007">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"> $M$-QR decomposition and hyperpower iterative methods for computing outer inverses of tensors </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Behera%2C+R">Ratikanta Behera</a>, <a href="/search/math?searchtype=author&query=Panigrahy%2C+K">Krushnachandra Panigrahy</a>, <a href="/search/math?searchtype=author&query=Sahoo%2C+J+K">Jajati Keshari Sahoo</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</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="2409.07007v1-abstract-short" style="display: inline;"> The outer inverse of tensors plays increasingly significant roles in computational mathematics, numerical analysis, and other generalized inverses of tensors. In this paper, we compute outer inverses with prescribed ranges and kernels of a given tensor through tensor QR decomposition and hyperpower iterative method under the M-product structure, which is a family of tensor-tensor products, general… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.07007v1-abstract-full').style.display = 'inline'; document.getElementById('2409.07007v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.07007v1-abstract-full" style="display: none;"> The outer inverse of tensors plays increasingly significant roles in computational mathematics, numerical analysis, and other generalized inverses of tensors. In this paper, we compute outer inverses with prescribed ranges and kernels of a given tensor through tensor QR decomposition and hyperpower iterative method under the M-product structure, which is a family of tensor-tensor products, generalization of the t-product and c-product, allows us to suit the physical interpretations across those different modes. We discuss a theoretical analysis of the nineteen-order convergence of the proposed tensor-based iterative method. Further, we design effective tensor-based algorithms for computing outer inverses using M-QR decomposition and hyperpower iterative method. The theoretical results are validated with numerical examples demonstrating the appropriateness of the proposed methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.07007v1-abstract-full').style.display = 'none'; document.getElementById('2409.07007v1-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> <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/2408.15586">arXiv:2408.15586</a> <span> [<a href="https://arxiv.org/pdf/2408.15586">pdf</a>, <a href="https://arxiv.org/ps/2408.15586">ps</a>, <a href="https://arxiv.org/format/2408.15586">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"> Decay Rates for Viscous Surface Waves of Isotropic Micropolar Fluids With or Without Surface Tension </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuchong Wei</a>, <a href="/search/math?searchtype=author&query=Yao%2C+Z">Zheng-an Yao</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="2408.15586v1-abstract-short" style="display: inline;"> In this paper, we consider a layer of viscous incompressible isotropic micropolar fluid in a uniform gravitational field of finite depth, lying above a flat rigid bottom and below the atmosphere in a three-dimensional horizontally periodic setting. The fluid dynamics are governed by gravity-driven incompressible micropolar equations. We investigate the global well-posedness for both the cases with… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.15586v1-abstract-full').style.display = 'inline'; document.getElementById('2408.15586v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.15586v1-abstract-full" style="display: none;"> In this paper, we consider a layer of viscous incompressible isotropic micropolar fluid in a uniform gravitational field of finite depth, lying above a flat rigid bottom and below the atmosphere in a three-dimensional horizontally periodic setting. The fluid dynamics are governed by gravity-driven incompressible micropolar equations. We investigate the global well-posedness for both the cases with and without surface tension. On one hand, in the case with surface tension (i.e. 蟽 > 0), we show that the global solution decays to the equilibrium exponentially. On the other hand, in the case without surface tension (i.e. 蟽 = 0), the solution decays to the equilibrium at an almost exponential rate. Comparing the two different cases for 蟽 > 0 and 蟽 = 0 reveals that the surface tension can enhance the decay rate. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.15586v1-abstract-full').style.display = 'none'; document.getElementById('2408.15586v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <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:1011.5179, arXiv:1911.06506 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/2408.02320">arXiv:2408.02320</a> <span> [<a href="https://arxiv.org/pdf/2408.02320">pdf</a>, <a href="https://arxiv.org/ps/2408.02320">ps</a>, <a href="https://arxiv.org/format/2408.02320">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="Signal Processing">eess.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <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 Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+G">Gen Li</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuting Wei</a>, <a href="/search/math?searchtype=author&query=Chi%2C+Y">Yuejie Chi</a>, <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yuxin Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.02320v1-abstract-short" style="display: inline;"> Diffusion models, which convert noise into new data instances by learning to reverse a diffusion process, have become a cornerstone in contemporary generative modeling. In this work, we develop non-asymptotic convergence theory for a popular diffusion-based sampler (i.e., the probability flow ODE sampler) in discrete time, assuming access to $\ell_2$-accurate estimates of the (Stein) score functio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02320v1-abstract-full').style.display = 'inline'; document.getElementById('2408.02320v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.02320v1-abstract-full" style="display: none;"> Diffusion models, which convert noise into new data instances by learning to reverse a diffusion process, have become a cornerstone in contemporary generative modeling. In this work, we develop non-asymptotic convergence theory for a popular diffusion-based sampler (i.e., the probability flow ODE sampler) in discrete time, assuming access to $\ell_2$-accurate estimates of the (Stein) score functions. For distributions in $\mathbb{R}^d$, we prove that $d/\varepsilon$ iterations -- modulo some logarithmic and lower-order terms -- are sufficient to approximate the target distribution to within $\varepsilon$ total-variation distance. This is the first result establishing nearly linear dimension-dependency (in $d$) for the probability flow ODE sampler. Imposing only minimal assumptions on the target data distribution (e.g., no smoothness assumption is imposed), our results also characterize how $\ell_2$ score estimation errors affect the quality of the data generation processes. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach without the need of resorting to SDE and ODE toolboxes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.02320v1-abstract-full').style.display = 'none'; document.getElementById('2408.02320v1-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">This manuscript presents improved theory for probability flow ODEs compared to its earlier version arXiv:2306.09251</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.18425">arXiv:2407.18425</a> <span> [<a href="https://arxiv.org/pdf/2407.18425">pdf</a>, <a href="https://arxiv.org/ps/2407.18425">ps</a>, <a href="https://arxiv.org/format/2407.18425">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"> Fujita phenomena in nonlinear fractional Rayleigh-Stokes equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+Y">Yiming Jiang</a>, <a href="/search/math?searchtype=author&query=Ren%2C+J">Jingchuang Ren</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yawei Wei</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="2407.18425v1-abstract-short" style="display: inline;"> This paper concerns the Cauchy problems for the nonlinear Rayleigh-Stokes equation and the corresponding system with time-fractional derivative of order $伪\in(0,1)$, which can be used to simulate the anomalous diffusion in viscoelastic fluids. It is shown that there exists the critical Fujita exponent which separates systematic blow-up of the solutions from possible global existence, and the criti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.18425v1-abstract-full').style.display = 'inline'; document.getElementById('2407.18425v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.18425v1-abstract-full" style="display: none;"> This paper concerns the Cauchy problems for the nonlinear Rayleigh-Stokes equation and the corresponding system with time-fractional derivative of order $伪\in(0,1)$, which can be used to simulate the anomalous diffusion in viscoelastic fluids. It is shown that there exists the critical Fujita exponent which separates systematic blow-up of the solutions from possible global existence, and the critical exponent is independent of the parameter $伪$. Different from the general scaling argument for parabolic problems, the main ingredients of our proof are suitable decay estimates of the solution operator and the construction of the test function. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.18425v1-abstract-full').style.display = 'none'; document.getElementById('2407.18425v1-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 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/2407.18401">arXiv:2407.18401</a> <span> [<a href="https://arxiv.org/pdf/2407.18401">pdf</a>, <a href="https://arxiv.org/ps/2407.18401">ps</a>, <a href="https://arxiv.org/format/2407.18401">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"> Stackelberg games with the third party </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+Y">Yiming Jiang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yawei Wei</a>, <a href="/search/math?searchtype=author&query=Xue%2C+J">Jie Xue</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="2407.18401v1-abstract-short" style="display: inline;"> In this paper, we introduce the third party to achieve the Stackelberg equilibrium with the time inconsistency in three different Stackelberg games, which are the discrete-time games, the dynamic games, and the mean field games. Here all followers are experiencing learning-by-doing. The role of a third party is similar to industry associations, they supervise the leader's implementation and impose… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.18401v1-abstract-full').style.display = 'inline'; document.getElementById('2407.18401v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.18401v1-abstract-full" style="display: none;"> In this paper, we introduce the third party to achieve the Stackelberg equilibrium with the time inconsistency in three different Stackelberg games, which are the discrete-time games, the dynamic games, and the mean field games. Here all followers are experiencing learning-by-doing. The role of a third party is similar to industry associations, they supervise the leader's implementation and impose penalties for the defection with the discount factor. Then we obtain different forms of discount factors in different models and effective conditions to prevent defection.These results are consistent and the third party intervention is effective and maneuverable in practice. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.18401v1-abstract-full').style.display = 'none'; document.getElementById('2407.18401v1-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 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/2407.18399">arXiv:2407.18399</a> <span> [<a href="https://arxiv.org/pdf/2407.18399">pdf</a>, <a href="https://arxiv.org/ps/2407.18399">ps</a>, <a href="https://arxiv.org/format/2407.18399">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> The kernel space of linear operator on a class of Grushin equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yawei Wei</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+X">Xiaodong 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="2407.18399v2-abstract-short" style="display: inline;"> In this paper, we concern the kernel of linear operator for a class of Grushin equation. First, we study the kernel space of linear operator for a general Grushin equation. Then, we provide an exact expression for the kernel space of linear operator for a special Grushin equation. Finally, we prove the linear operator related to the singularly perturbed Grushin equation is invertible when restrict… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.18399v2-abstract-full').style.display = 'inline'; document.getElementById('2407.18399v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.18399v2-abstract-full" style="display: none;"> In this paper, we concern the kernel of linear operator for a class of Grushin equation. First, we study the kernel space of linear operator for a general Grushin equation. Then, we provide an exact expression for the kernel space of linear operator for a special Grushin equation. Finally, we prove the linear operator related to the singularly perturbed Grushin equation is invertible when restricted to the complement of its approximate kernel space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.18399v2-abstract-full').style.display = 'none'; document.getElementById('2407.18399v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 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">MSC Class:</span> 35B40; 47B38; 35J70 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.05159">arXiv:2406.05159</a> <span> [<a href="https://arxiv.org/pdf/2406.05159">pdf</a>, <a href="https://arxiv.org/ps/2406.05159">ps</a>, <a href="https://arxiv.org/format/2406.05159">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yong Wei</a>, <a href="/search/math?searchtype=author&query=Yang%2C+B">Bo Yang</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+T">Tailong 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="2406.05159v1-abstract-short" style="display: inline;"> We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$ with speed given by a general nonhomogeneous function of the Gauss curvature. For a large class of speed functions, we prove that the solution of the flow remains convex, exists for all positive time $t\in [0,\infty)$ and converges to a geodesic sphere exponentially as… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.05159v1-abstract-full').style.display = 'inline'; document.getElementById('2406.05159v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.05159v1-abstract-full" style="display: none;"> We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$ with speed given by a general nonhomogeneous function of the Gauss curvature. For a large class of speed functions, we prove that the solution of the flow remains convex, exists for all positive time $t\in [0,\infty)$ and converges to a geodesic sphere exponentially as $t\to\infty$ in the smooth topology. A key step is to show the $L^1$ oscillation decay of the Gauss curvature to its average along a subsequence of times going to the infinity, which combined with an argument using the hyperbolic curvature measure theory implies the Hausdorff convergence. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.05159v1-abstract-full').style.display = 'none'; document.getElementById('2406.05159v1-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 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">27 pages. All comments are welcome. arXiv admin note: substantial text overlap with arXiv:2210.06035</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53E10; 53C42 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.08639">arXiv:2405.08639</a> <span> [<a href="https://arxiv.org/pdf/2405.08639">pdf</a>, <a href="https://arxiv.org/ps/2405.08639">ps</a>, <a href="https://arxiv.org/format/2405.08639">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Logic">math.LO</span> </div> </div> <p class="title is-5 mathjax"> Upwards homogeneity in iterated symmetric extensions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ryan-Smith%2C+C">Calliope Ryan-Smith</a>, <a href="/search/math?searchtype=author&query=Schilhan%2C+J">Jonathan Schilhan</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yujun Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.08639v1-abstract-short" style="display: inline;"> It is sometimes desirable in choiceless constructions of set theory that one iteratively extends some ground model without adding new sets of ordinals after the first extension. Pushing this further, one may wish to have models $V \subseteq M \subseteq N$ of $\mathsf{ZF}$ such that $N$ contains no subsets of $V$ that do not already appear in $M$. We isolate, in the case that $M$ and $N$ are symmet… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.08639v1-abstract-full').style.display = 'inline'; document.getElementById('2405.08639v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.08639v1-abstract-full" style="display: none;"> It is sometimes desirable in choiceless constructions of set theory that one iteratively extends some ground model without adding new sets of ordinals after the first extension. Pushing this further, one may wish to have models $V \subseteq M \subseteq N$ of $\mathsf{ZF}$ such that $N$ contains no subsets of $V$ that do not already appear in $M$. We isolate, in the case that $M$ and $N$ are symmetric extensions (particular inner models of a generic extension of $V$), the exact conditions that cause this behaviour and show how it can broadly be applied to many known constructions. We call this behaviour upwards homogeneity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.08639v1-abstract-full').style.display = 'none'; document.getElementById('2405.08639v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">16 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 03E25 (Primary) 03E35; 03E40 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.07147">arXiv:2405.07147</a> <span> [<a href="https://arxiv.org/pdf/2405.07147">pdf</a>, <a href="https://arxiv.org/ps/2405.07147">ps</a>, <a href="https://arxiv.org/format/2405.07147">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"> Randomized algorithms for computing the tensor train approximation and their applications </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Che%2C+M">Maolin Che</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</a>, <a href="/search/math?searchtype=author&query=Yan%2C+H">Hong Yan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.07147v2-abstract-short" style="display: inline;"> In this paper, we focus on the fixed TT-rank and precision problems of finding an approximation of the tensor train (TT) decomposition of a tensor. Note that the TT-SVD and TT-cross are two well-known algorithms for these two problems. Firstly, by combining the random projection technique with the power scheme, we obtain two types of randomized algorithms for the fixed TT-rank problem. Secondly, b… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.07147v2-abstract-full').style.display = 'inline'; document.getElementById('2405.07147v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.07147v2-abstract-full" style="display: none;"> In this paper, we focus on the fixed TT-rank and precision problems of finding an approximation of the tensor train (TT) decomposition of a tensor. Note that the TT-SVD and TT-cross are two well-known algorithms for these two problems. Firstly, by combining the random projection technique with the power scheme, we obtain two types of randomized algorithms for the fixed TT-rank problem. Secondly, by using the non-asymptotic theory of sub-random Gaussian matrices, we derive the upper bounds of the proposed randomized algorithms. Thirdly, we deduce a new deterministic strategy to estimate the desired TT-rank with a given tolerance and another adaptive randomized algorithm that finds a low TT-rank representation satisfying a given tolerance, and is beneficial when the target TT-rank is not known in advance. We finally illustrate the accuracy of the proposed algorithms via some test tensors from synthetic and real databases. In particular, for the fixed TT-rank problem, the proposed algorithms can be several times faster than the TT-SVD, and the accuracy of the proposed algorithms and the TT-SVD are comparable for several test tensors. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.07147v2-abstract-full').style.display = 'none'; document.getElementById('2405.07147v2-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 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">40 pages, 9 figures and 4 tables</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 15A18; 15A69; 65F55; 68W20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.13525">arXiv:2404.13525</a> <span> [<a href="https://arxiv.org/pdf/2404.13525">pdf</a>, <a href="https://arxiv.org/format/2404.13525">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"> QR Decomposition of Dual Matrices and its Application to Traveling Wave Identification in the Brain </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xu%2C+R">Renjie Xu</a>, <a href="/search/math?searchtype=author&query=Wei%2C+T">Tong Wei</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</a>, <a href="/search/math?searchtype=author&query=Xie%2C+P">Pengpeng Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.13525v1-abstract-short" style="display: inline;"> Matrix decompositions in dual number representations have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices, specifically geared towards its application in traveling wave identification, utilizing the concept of proper orthogonal decomposition. When dealing with large-scale pr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.13525v1-abstract-full').style.display = 'inline'; document.getElementById('2404.13525v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.13525v1-abstract-full" style="display: none;"> Matrix decompositions in dual number representations have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices, specifically geared towards its application in traveling wave identification, utilizing the concept of proper orthogonal decomposition. When dealing with large-scale problems, we present explicit solutions for the QR, thin QR, and randomized QR decompositions of dual number matrices, along with their respective algorithms with column pivoting. The QR decomposition of dual matrices is an accurate first-order perturbation, with the Q-factor satisfying rigorous perturbation bounds, leading to enhanced orthogonality. In numerical experiments, we discuss the suitability of different QR algorithms when confronted with various large-scale dual matrices, providing their respective domains of applicability. Subsequently, we employed the QR decomposition of dual matrices to compute the DMPGI, thereby attaining results of higher precision. Moreover, we apply the QR decomposition in the context of traveling wave identification, employing the notion of proper orthogonal decomposition to perform a validation analysis of large-scale functional magnetic resonance imaging (fMRI) data for brain functional circuits. Our approach significantly improves the identification of two types of wave signals compared to previous research, providing empirical evidence for cognitive neuroscience theories. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.13525v1-abstract-full').style.display = 'none'; document.getElementById('2404.13525v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.11991">arXiv:2404.11991</a> <span> [<a href="https://arxiv.org/pdf/2404.11991">pdf</a>, <a href="https://arxiv.org/ps/2404.11991">ps</a>, <a href="https://arxiv.org/format/2404.11991">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"> Pohozaev identities and Kelvin transformation of semilinear Grushin equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yawei Wei</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+X">Xiaodong 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="2404.11991v1-abstract-short" style="display: inline;"> In this paper, we study Pohozaev identities, Kelvin transformation and their applications of semilinear Grushin equation. First, we establish two Pohozaev identities generated from translations and determine the location of the concentration point for solution of a kind of Grushin equation by such identities. Next, we establish Pohozaev identity generated from scaling and prove the nonexistence of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.11991v1-abstract-full').style.display = 'inline'; document.getElementById('2404.11991v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.11991v1-abstract-full" style="display: none;"> In this paper, we study Pohozaev identities, Kelvin transformation and their applications of semilinear Grushin equation. First, we establish two Pohozaev identities generated from translations and determine the location of the concentration point for solution of a kind of Grushin equation by such identities. Next, we establish Pohozaev identity generated from scaling and prove the nonexistence of nontrivial solutions of another kind of Grushin equation by such identity. Finally, we provide the change of Grushin operator by Kelvin transformation and obtain the decay rate of solution at infinity for a critical Grushin equation by Kelvin transformation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.11991v1-abstract-full').style.display = 'none'; document.getElementById('2404.11991v1-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35J70; 35A22; 35B40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.10298">arXiv:2404.10298</a> <span> [<a href="https://arxiv.org/pdf/2404.10298">pdf</a>, <a href="https://arxiv.org/ps/2404.10298">ps</a>, <a href="https://arxiv.org/format/2404.10298">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Anisotropic Gauss curvature flow of complete non-compact graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Pan%2C+S">Shujing Pan</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yong Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.10298v1-abstract-short" style="display: inline;"> In this paper, we consider the anisotropic $伪$-Gauss curvature flow for complete noncompact convex hypersurfaces in the Euclidean space with the anisotropy determined by a smooth closed uniformly convex Wulff shape. We show that for all positive power $伪>0$, if the initial hypersurface is complete noncompact and locally uniformly convex, then the solution of the flow exists for all positive time. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.10298v1-abstract-full" style="display: none;"> In this paper, we consider the anisotropic $伪$-Gauss curvature flow for complete noncompact convex hypersurfaces in the Euclidean space with the anisotropy determined by a smooth closed uniformly convex Wulff shape. We show that for all positive power $伪>0$, if the initial hypersurface is complete noncompact and locally uniformly convex, then the solution of the flow exists for all positive time. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.10298v1-abstract-full').style.display = 'none'; document.getElementById('2404.10298v1-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages. All comments are welcome</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C44; 53C42 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.08196">arXiv:2404.08196</a> <span> [<a href="https://arxiv.org/pdf/2404.08196">pdf</a>, <a href="https://arxiv.org/ps/2404.08196">ps</a>, <a href="https://arxiv.org/format/2404.08196">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"> Properties of fractional p-Laplace equations with sign-changing potential </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Duan%2C+Y">Yubo Duan</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yawei Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.08196v1-abstract-short" style="display: inline;"> In this paper, we consider the nonlinear equation involving the fractional p-Laplacian with sign-changing potential. This model draws inspiration from De Giorgi Conjecture. There are two main results in this paper. Firstly, we obtain that the solution is radially symmetric within the bounded domain, by applying the moving plane method. Secondly, by exploiting the idea of the sliding method, we con… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.08196v1-abstract-full').style.display = 'inline'; document.getElementById('2404.08196v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.08196v1-abstract-full" style="display: none;"> In this paper, we consider the nonlinear equation involving the fractional p-Laplacian with sign-changing potential. This model draws inspiration from De Giorgi Conjecture. There are two main results in this paper. Firstly, we obtain that the solution is radially symmetric within the bounded domain, by applying the moving plane method. Secondly, by exploiting the idea of the sliding method, we construct the appropriate auxiliary functions to prove that the solution is monotone increasing in some direction in the unbounded domain. The different properties of the solution in bounded and unbounded domains are mainly attributed to the inherent non-locality of the fractional p-Laplacian. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.08196v1-abstract-full').style.display = 'none'; document.getElementById('2404.08196v1-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R11; 35J60; 35B40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.08192">arXiv:2404.08192</a> <span> [<a href="https://arxiv.org/pdf/2404.08192">pdf</a>, <a href="https://arxiv.org/ps/2404.08192">ps</a>, <a href="https://arxiv.org/format/2404.08192">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"> Wellposedness of the Master Equation for Mean Field Games with Grushin Type Diffusion </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+Y">Yiming Jiang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yawei Wei</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yiyun 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="2404.08192v1-abstract-short" style="display: inline;"> We study the wellposedness of the master equation for a second-order mean field games with the Grushin type diffusion. In order to do this, we obtain the properties of its solution by investigating a degenerate mean field games system for which there exists an equivalent characterization with the master equation. The crucial points of this paper are to explore some regularities of solutions to two… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.08192v1-abstract-full').style.display = 'inline'; document.getElementById('2404.08192v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.08192v1-abstract-full" style="display: none;"> We study the wellposedness of the master equation for a second-order mean field games with the Grushin type diffusion. In order to do this, we obtain the properties of its solution by investigating a degenerate mean field games system for which there exists an equivalent characterization with the master equation. The crucial points of this paper are to explore some regularities of solutions to two types of linear degenerate partial differential equations and a kind of degenerate linear coupled system so as to derive the existence of solutions to the master equation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.08192v1-abstract-full').style.display = 'none'; document.getElementById('2404.08192v1-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.06372">arXiv:2404.06372</a> <span> [<a href="https://arxiv.org/pdf/2404.06372">pdf</a>, <a href="https://arxiv.org/ps/2404.06372">ps</a>, <a href="https://arxiv.org/format/2404.06372">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"> Existence and uniqueness for cone degenerate p-Laplace equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+H">Hua Chen</a>, <a href="/search/math?searchtype=author&query=Hu%2C+J">Jiangtao Hu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+X">Xiaochun Liu</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yawei Wei</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+M">Mengnan Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.06372v2-abstract-short" style="display: inline;"> In this paper, we study the cone degenerate p-Laplace equation. We provide the existence of the viscosity solutions by proving Alexandrov-Bakelman-Pucci and H枚lder estimates. Further more, we give the comparison principle by an equivalent transformation. Finally, we obtain the existence of weak solutions by analyzing the relationship between weak solutions and viscosity solutions. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.06372v2-abstract-full" style="display: none;"> In this paper, we study the cone degenerate p-Laplace equation. We provide the existence of the viscosity solutions by proving Alexandrov-Bakelman-Pucci and H枚lder estimates. Further more, we give the comparison principle by an equivalent transformation. Finally, we obtain the existence of weak solutions by analyzing the relationship between weak solutions and viscosity solutions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.06372v2-abstract-full').style.display = 'none'; document.getElementById('2404.06372v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.05251">arXiv:2404.05251</a> <span> [<a href="https://arxiv.org/pdf/2404.05251">pdf</a>, <a href="https://arxiv.org/ps/2404.05251">ps</a>, <a href="https://arxiv.org/format/2404.05251">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> Association schemes arising from non-weakly regular bent functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yadi Wei</a>, <a href="/search/math?searchtype=author&query=Wang%2C+J">Jiaxin Wang</a>, <a href="/search/math?searchtype=author&query=Fu%2C+F">Fang-Wei Fu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.05251v2-abstract-short" style="display: inline;"> Association schemes play an important role in algebraic combinatorics and have important applications in coding theory, graph theory and design theory. The methods to construct association schemes by using bent functions have been extensively studied. Recently, in [13], {脰}zbudak and Pelen constructed infinite families of symmetric association schemes of classes $5$ and $6$ by using ternary non-we… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.05251v2-abstract-full').style.display = 'inline'; document.getElementById('2404.05251v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.05251v2-abstract-full" style="display: none;"> Association schemes play an important role in algebraic combinatorics and have important applications in coding theory, graph theory and design theory. The methods to construct association schemes by using bent functions have been extensively studied. Recently, in [13], {脰}zbudak and Pelen constructed infinite families of symmetric association schemes of classes $5$ and $6$ by using ternary non-weakly regular bent functions.They also stated that constructing $2p$-class association schemes from $p$-ary non-weakly regular bent functions is an interesting problem, where $p>3$ is an odd prime. In this paper, using non-weakly regular bent functions, we construct infinite families of symmetric association schemes of classes $2p$, $(2p+1)$ and $\frac{3p+1}{2}$ for any odd prime $p$. Fusing those association schemes, we also obtain $t$-class symmetric association schemes, where $t=4,5,6,7$. In addition, we give the sufficient and necessary conditions for the partitions $P$, $D$, $T$, $U$ and $V$ (defined in this paper) to induce symmetric association schemes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.05251v2-abstract-full').style.display = 'none'; document.getElementById('2404.05251v2-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.04062">arXiv:2404.04062</a> <span> [<a href="https://arxiv.org/pdf/2404.04062">pdf</a>, <a href="https://arxiv.org/format/2404.04062">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"> Derivative-free tree optimization for complex systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Ye Wei</a>, <a href="/search/math?searchtype=author&query=Peng%2C+B">Bo Peng</a>, <a href="/search/math?searchtype=author&query=Xie%2C+R">Ruiwen Xie</a>, <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yangtao Chen</a>, <a href="/search/math?searchtype=author&query=Qin%2C+Y">Yu Qin</a>, <a href="/search/math?searchtype=author&query=Wen%2C+P">Peng Wen</a>, <a href="/search/math?searchtype=author&query=Bauer%2C+S">Stefan Bauer</a>, <a href="/search/math?searchtype=author&query=Tung%2C+P">Po-Yen Tung</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.04062v1-abstract-short" style="display: inline;"> A tremendous range of design tasks in materials, physics, and biology can be formulated as finding the optimum of an objective function depending on many parameters without knowing its closed-form expression or the derivative. Traditional derivative-free optimization techniques often rely on strong assumptions about objective functions, thereby failing at optimizing non-convex systems beyond 100 d… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.04062v1-abstract-full').style.display = 'inline'; document.getElementById('2404.04062v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.04062v1-abstract-full" style="display: none;"> A tremendous range of design tasks in materials, physics, and biology can be formulated as finding the optimum of an objective function depending on many parameters without knowing its closed-form expression or the derivative. Traditional derivative-free optimization techniques often rely on strong assumptions about objective functions, thereby failing at optimizing non-convex systems beyond 100 dimensions. Here, we present a tree search method for derivative-free optimization that enables accelerated optimal design of high-dimensional complex systems. Specifically, we introduce stochastic tree expansion, dynamic upper confidence bound, and short-range backpropagation mechanism to evade local optimum, iteratively approximating the global optimum using machine learning models. This development effectively confronts the dimensionally challenging problems, achieving convergence to global optima across various benchmark functions up to 2,000 dimensions, surpassing the existing methods by 10- to 20-fold. Our method demonstrates wide applicability to a wide range of real-world complex systems spanning materials, physics, and biology, considerably outperforming state-of-the-art algorithms. This enables efficient autonomous knowledge discovery and facilitates self-driving virtual laboratories. Although we focus on problems within the realm of natural science, the advancements in optimization techniques achieved herein are applicable to a broader spectrum of challenges across all quantitative disciplines. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.04062v1-abstract-full').style.display = 'none'; document.getElementById('2404.04062v1-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">39 pages, 3 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/2403.04892">arXiv:2403.04892</a> <span> [<a href="https://arxiv.org/pdf/2403.04892">pdf</a>, <a href="https://arxiv.org/format/2403.04892">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Operator Algebras">math.OA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Generalized Choi-Davis-Jensen's Operator Inequalities and Their Applications </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chang%2C+S+Y">Shih Yu Chang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.04892v1-abstract-short" style="display: inline;"> The original Choi-Davis-Jensen's inequality, with its wide-ranging applications in diverse scientific and engineering fields, has motivated researchers to explore generalizations. In this study, we extend Davis-Choi-Jensen's inequality by considering a nonlinear map instead of a normalized linear map and generalize operator convex function to any continuous function defined in a compact region. Th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.04892v1-abstract-full').style.display = 'inline'; document.getElementById('2403.04892v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.04892v1-abstract-full" style="display: none;"> The original Choi-Davis-Jensen's inequality, with its wide-ranging applications in diverse scientific and engineering fields, has motivated researchers to explore generalizations. In this study, we extend Davis-Choi-Jensen's inequality by considering a nonlinear map instead of a normalized linear map and generalize operator convex function to any continuous function defined in a compact region. The Stone-Weierstrass theorem and Kantorovich function are instrumental in formulating and proving generalized Choi-Davis-Jensen's inequalities. Additionally, we present an application of this generalized inequality in the context of statistical physics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.04892v1-abstract-full').style.display = 'none'; document.getElementById('2403.04892v1-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 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.03852">arXiv:2403.03852</a> <span> [<a href="https://arxiv.org/pdf/2403.03852">pdf</a>, <a href="https://arxiv.org/format/2403.03852">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</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="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Accelerating Convergence of Score-Based Diffusion Models, Provably </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+G">Gen Li</a>, <a href="/search/math?searchtype=author&query=Huang%2C+Y">Yu Huang</a>, <a href="/search/math?searchtype=author&query=Efimov%2C+T">Timofey Efimov</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuting Wei</a>, <a href="/search/math?searchtype=author&query=Chi%2C+Y">Yuejie Chi</a>, <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yuxin Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.03852v1-abstract-short" style="display: inline;"> Score-based diffusion models, while achieving remarkable empirical performance, often suffer from low sampling speed, due to extensive function evaluations needed during the sampling phase. Despite a flurry of recent activities towards speeding up diffusion generative modeling in practice, theoretical underpinnings for acceleration techniques remain severely limited. In this paper, we design novel… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.03852v1-abstract-full').style.display = 'inline'; document.getElementById('2403.03852v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.03852v1-abstract-full" style="display: none;"> Score-based diffusion models, while achieving remarkable empirical performance, often suffer from low sampling speed, due to extensive function evaluations needed during the sampling phase. Despite a flurry of recent activities towards speeding up diffusion generative modeling in practice, theoretical underpinnings for acceleration techniques remain severely limited. In this paper, we design novel training-free algorithms to accelerate popular deterministic (i.e., DDIM) and stochastic (i.e., DDPM) samplers. Our accelerated deterministic sampler converges at a rate $O(1/{T}^2)$ with $T$ the number of steps, improving upon the $O(1/T)$ rate for the DDIM sampler; and our accelerated stochastic sampler converges at a rate $O(1/T)$, outperforming the rate $O(1/\sqrt{T})$ for the DDPM sampler. The design of our algorithms leverages insights from higher-order approximation, and shares similar intuitions as popular high-order ODE solvers like the DPM-Solver-2. Our theory accommodates $\ell_2$-accurate score estimates, and does not require log-concavity or smoothness on the target distribution. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.03852v1-abstract-full').style.display = 'none'; document.getElementById('2403.03852v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">The first two authors contributed equally</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.16213">arXiv:2402.16213</a> <span> [<a href="https://arxiv.org/pdf/2402.16213">pdf</a>, <a href="https://arxiv.org/ps/2402.16213">ps</a>, <a href="https://arxiv.org/format/2402.16213">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="Classical Analysis and ODEs">math.CA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jde.2024.08.048">10.1016/j.jde.2024.08.048 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Sparse gradient bounds for divergence form elliptic equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Saari%2C+O">Olli Saari</a>, <a href="/search/math?searchtype=author&query=Wang%2C+H">Hua-Yang Wang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuanhong Wei</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="2402.16213v2-abstract-short" style="display: inline;"> We provide sparse estimates for gradients of solutions to divergence form elliptic partial differential equations in terms of the source data. We give a general result of Meyers (or Gehring) type, a result for linear equations with VMO coefficients and a result for linear equations with Dini continuous coefficients. In addition, we provide an abstract theorem conditional on PDE estimates available… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.16213v2-abstract-full').style.display = 'inline'; document.getElementById('2402.16213v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.16213v2-abstract-full" style="display: none;"> We provide sparse estimates for gradients of solutions to divergence form elliptic partial differential equations in terms of the source data. We give a general result of Meyers (or Gehring) type, a result for linear equations with VMO coefficients and a result for linear equations with Dini continuous coefficients. In addition, we provide an abstract theorem conditional on PDE estimates available. The linear results have the full range of weighted estimates with Muckenhoupt weights as a consequence. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.16213v2-abstract-full').style.display = 'none'; document.getElementById('2402.16213v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">v2: writing improved all over and more details added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Differential Equations 413 (2024), 606-631 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.13491">arXiv:2402.13491</a> <span> [<a href="https://arxiv.org/pdf/2402.13491">pdf</a>, <a href="https://arxiv.org/format/2402.13491">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"> Algebraic Riccati Tensor Equations with Applications in Multilinear Control Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yuchao Wang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+G">Guofeng Zhang</a>, <a href="/search/math?searchtype=author&query=Chang%2C+S+Y">Shih Yu Chang</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="2402.13491v2-abstract-short" style="display: inline;"> In a recent paper by Chen et al. [9], the authors initiated the control-theoretic study of a class of discrete-time multilinear time-invariant (MLTI) control systems, where system states, inputs, and outputs are all tensors endowed with the Einstein product. They established criteria for fundamental system-theoretic notions such as stability, reachability, and observability through tensor decompos… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.13491v2-abstract-full').style.display = 'inline'; document.getElementById('2402.13491v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.13491v2-abstract-full" style="display: none;"> In a recent paper by Chen et al. [9], the authors initiated the control-theoretic study of a class of discrete-time multilinear time-invariant (MLTI) control systems, where system states, inputs, and outputs are all tensors endowed with the Einstein product. They established criteria for fundamental system-theoretic notions such as stability, reachability, and observability through tensor decomposition. Building on this new research direction, the purpose of our paper is to extend the study to continuous-time MLTI control systems. Specifically, we define Hamiltonian tensors and symplectic tensors, and we establish the Schur-Hamiltonian tensor decomposition and the symplectic tensor singular value decomposition (SVD). Based on these concepts, we propose the algebraic Riccati tensor equation (ARTE) and demonstrate that it has a unique positive semidefinite solution if the system is stabilizable and detectable. To find numerical solutions to the ARTE, we introduce a tensor-based Newton method. Additionally, we establish the tensor versions of the bounded real lemma and the small gain theorem. A first-order robustness analysis of the ARTE is also conducted. Finally, we provide a numerical example to illustrate the proposed theory and algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.13491v2-abstract-full').style.display = 'none'; document.getElementById('2402.13491v2-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">v1</span> submitted 20 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages, 7 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 15A69; 93B35; 93C05; 93D15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.07802">arXiv:2402.07802</a> <span> [<a href="https://arxiv.org/pdf/2402.07802">pdf</a>, <a href="https://arxiv.org/ps/2402.07802">ps</a>, <a href="https://arxiv.org/format/2402.07802">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"> Towards a mathematical theory for consistency training in diffusion models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+G">Gen Li</a>, <a href="/search/math?searchtype=author&query=Huang%2C+Z">Zhihan Huang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuting Wei</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="2402.07802v1-abstract-short" style="display: inline;"> Consistency models, which were proposed to mitigate the high computational overhead during the sampling phase of diffusion models, facilitate single-step sampling while attaining state-of-the-art empirical performance. When integrated into the training phase, consistency models attempt to train a sequence of consistency functions capable of mapping any point at any time step of the diffusion proce… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.07802v1-abstract-full').style.display = 'inline'; document.getElementById('2402.07802v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.07802v1-abstract-full" style="display: none;"> Consistency models, which were proposed to mitigate the high computational overhead during the sampling phase of diffusion models, facilitate single-step sampling while attaining state-of-the-art empirical performance. When integrated into the training phase, consistency models attempt to train a sequence of consistency functions capable of mapping any point at any time step of the diffusion process to its starting point. Despite the empirical success, a comprehensive theoretical understanding of consistency training remains elusive. This paper takes a first step towards establishing theoretical underpinnings for consistency models. We demonstrate that, in order to generate samples within $\varepsilon$ proximity to the target in distribution (measured by some Wasserstein metric), it suffices for the number of steps in consistency learning to exceed the order of $d^{5/2}/\varepsilon$, with $d$ the data dimension. Our theory offers rigorous insights into the validity and efficacy of consistency models, illuminating their utility in downstream inference tasks. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.07802v1-abstract-full').style.display = 'none'; document.getElementById('2402.07802v1-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 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">The first two authors contributed equally</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.16836">arXiv:2401.16836</a> <span> [<a href="https://arxiv.org/pdf/2401.16836">pdf</a>, <a href="https://arxiv.org/format/2401.16836">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"> Coseparable Nonnegative Tensor Factorization With T-CUR Decomposition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+J">Juefei Chen</a>, <a href="/search/math?searchtype=author&query=Huang%2C+L">Longxiu Huang</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.16836v3-abstract-short" style="display: inline;"> Nonnegative Matrix Factorization (NMF) is an important unsupervised learning method to extract meaningful features from data. To address the NMF problem within a polynomial time framework, researchers have introduced a separability assumption, which has recently evolved into the concept of coseparability. This advancement offers a more efficient core representation for the original data. However,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.16836v3-abstract-full').style.display = 'inline'; document.getElementById('2401.16836v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.16836v3-abstract-full" style="display: none;"> Nonnegative Matrix Factorization (NMF) is an important unsupervised learning method to extract meaningful features from data. To address the NMF problem within a polynomial time framework, researchers have introduced a separability assumption, which has recently evolved into the concept of coseparability. This advancement offers a more efficient core representation for the original data. However, in the real world, the data is more natural to be represented as a multi-dimensional array, such as images or videos. The NMF's application to high-dimensional data involves vectorization, which risks losing essential multi-dimensional correlations. To retain these inherent correlations in the data, we turn to tensors (multidimensional arrays) and leverage the tensor t-product. This approach extends the coseparable NMF to the tensor setting, creating what we term coseparable Nonnegative Tensor Factorization (NTF). In this work, we provide an alternating index selection method to select the coseparable core. Furthermore, we validate the t-CUR sampling theory and integrate it with the tensor Discrete Empirical Interpolation Method (t-DEIM) to introduce an alternative, randomized index selection process. These methods have been tested on both synthetic and facial analysis datasets. The results demonstrate the efficiency of coseparable NTF when compared to coseparable NMF. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.16836v3-abstract-full').style.display = 'none'; document.getElementById('2401.16836v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.13933">arXiv:2401.13933</a> <span> [<a href="https://arxiv.org/pdf/2401.13933">pdf</a>, <a href="https://arxiv.org/ps/2401.13933">ps</a>, <a href="https://arxiv.org/format/2401.13933">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"> Solutions to the First Order Difference Equations in the Multivariate Difference Field </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Du%2C+L">Lixin Du</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yarong Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.13933v1-abstract-short" style="display: inline;"> The bivariate difference field provides an algebraic framework for a sequence satisfying a recurrence of order two. Based on this, we focus on sequences satisfying a recurrence of higher order, and consider the multivariate difference field, in which the summation problem could be transformed into solving the first order difference equations. We then show a criterion for deciding whether the diffe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.13933v1-abstract-full').style.display = 'inline'; document.getElementById('2401.13933v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.13933v1-abstract-full" style="display: none;"> The bivariate difference field provides an algebraic framework for a sequence satisfying a recurrence of order two. Based on this, we focus on sequences satisfying a recurrence of higher order, and consider the multivariate difference field, in which the summation problem could be transformed into solving the first order difference equations. We then show a criterion for deciding whether the difference equation has a rational solution and present an algorithm for computing one rational solution of such a difference equation, if it exists. Moreover we get the rational solution set of such an equation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.13933v1-abstract-full').style.display = 'none'; document.getElementById('2401.13933v1-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> 24 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.11388">arXiv:2401.11388</a> <span> [<a href="https://arxiv.org/pdf/2401.11388">pdf</a>, <a href="https://arxiv.org/ps/2401.11388">ps</a>, <a href="https://arxiv.org/format/2401.11388">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"> Polynomial Solutions to the First Order Difference Equations in the Bivariate Difference Field </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yarong Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.11388v1-abstract-short" style="display: inline;"> The bivariate difference filed $(\mathbb{F}(伪, 尾), 蟽)$ provides an algebraic framework for a sequence satisfying a recurrence of order two and it could transform the summation involving a sequence satisfying a recurrence of order two into the first order difference equations in the bivariate difference field. Based on it, we present an algorithm for finding all the polynomial solutions of such equ… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.11388v1-abstract-full').style.display = 'inline'; document.getElementById('2401.11388v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.11388v1-abstract-full" style="display: none;"> The bivariate difference filed $(\mathbb{F}(伪, 尾), 蟽)$ provides an algebraic framework for a sequence satisfying a recurrence of order two and it could transform the summation involving a sequence satisfying a recurrence of order two into the first order difference equations in the bivariate difference field. Based on it, we present an algorithm for finding all the polynomial solutions of such equations in the bivariate difference field, and show an upper bound on the degree for polynomial solutions which is sufficient to compute polynomial solution by using the undetermined method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.11388v1-abstract-full').style.display = 'none'; document.getElementById('2401.11388v1-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 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.11387">arXiv:2401.11387</a> <span> [<a href="https://arxiv.org/pdf/2401.11387">pdf</a>, <a href="https://arxiv.org/ps/2401.11387">ps</a>, <a href="https://arxiv.org/format/2401.11387">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"> Rational Solutions to the First Order Difference Equations in the Bivariate Difference Field </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hou%2C+Q">Qing-Hu Hou</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yarong Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.11387v1-abstract-short" style="display: inline;"> Inspired by Karr's algorithm, we consider the summations involving a sequence satisfying a recurrence of order two. The structure of such summations provides an algebraic framework for solving the difference equations of form $a蟽(g)+bg=f$ in the bivariate difference field $(\mathbb{F}(伪, 尾), 蟽)$, where $a, b,f\in\mathbb{F}(伪,尾)\setminus\{0\}$ are known binary functions of $伪$, $尾$, and $伪$, $尾$ ar… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.11387v1-abstract-full').style.display = 'inline'; document.getElementById('2401.11387v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.11387v1-abstract-full" style="display: none;"> Inspired by Karr's algorithm, we consider the summations involving a sequence satisfying a recurrence of order two. The structure of such summations provides an algebraic framework for solving the difference equations of form $a蟽(g)+bg=f$ in the bivariate difference field $(\mathbb{F}(伪, 尾), 蟽)$, where $a, b,f\in\mathbb{F}(伪,尾)\setminus\{0\}$ are known binary functions of $伪$, $尾$, and $伪$, $尾$ are two algebraically independent transcendental elements, $蟽$ is a transformation that satisfies $蟽(伪)=尾$, $蟽(尾)=u伪+v尾$, where $u,v\neq 0\in\mathbb{F}$. Based on it, we then describe algorithms for finding the universal denominator for those equations in the bivariate difference field under certain assumptions. This reduces the general problem of finding the rational solutions of such equations to the problem of finding the polynomial solutions of such equations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.11387v1-abstract-full').style.display = 'none'; document.getElementById('2401.11387v1-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 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.03923">arXiv:2401.03923</a> <span> [<a href="https://arxiv.org/pdf/2401.03923">pdf</a>, <a href="https://arxiv.org/format/2401.03923">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Signal Processing">eess.SP</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 non-asymptotic distributional theory of approximate message passing for sparse and robust regression </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+G">Gen Li</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuting Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.03923v1-abstract-short" style="display: inline;"> Characterizing the distribution of high-dimensional statistical estimators is a challenging task, due to the breakdown of classical asymptotic theory in high dimension. This paper makes progress towards this by developing non-asymptotic distributional characterizations for approximate message passing (AMP) -- a family of iterative algorithms that prove effective as both fast estimators and powerfu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.03923v1-abstract-full').style.display = 'inline'; document.getElementById('2401.03923v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.03923v1-abstract-full" style="display: none;"> Characterizing the distribution of high-dimensional statistical estimators is a challenging task, due to the breakdown of classical asymptotic theory in high dimension. This paper makes progress towards this by developing non-asymptotic distributional characterizations for approximate message passing (AMP) -- a family of iterative algorithms that prove effective as both fast estimators and powerful theoretical machinery -- for both sparse and robust regression. Prior AMP theory, which focused on high-dimensional asymptotics for the most part, failed to describe the behavior of AMP when the number of iterations exceeds $o\big({\log n}/{\log \log n}\big)$ (with $n$ the sample size). We establish the first finite-sample non-asymptotic distributional theory of AMP for both sparse and robust regression that accommodates a polynomial number of iterations. Our results derive approximate accuracy of Gaussian approximation of the AMP iterates, which improves upon all prior results and implies enhanced distributional characterizations for both optimally tuned Lasso and robust M-estimator. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.03923v1-abstract-full').style.display = 'none'; document.getElementById('2401.03923v1-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 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.01423">arXiv:2401.01423</a> <span> [<a href="https://arxiv.org/pdf/2401.01423">pdf</a>, <a href="https://arxiv.org/format/2401.01423">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Hadamard integrators for wave equations in time and frequency domain: Eulerian formulations via butterfly algorithms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuxiao Wei</a>, <a href="/search/math?searchtype=author&query=Cheng%2C+J">Jin Cheng</a>, <a href="/search/math?searchtype=author&query=Leung%2C+S">Shingyu Leung</a>, <a href="/search/math?searchtype=author&query=Burridge%2C+R">Robert Burridge</a>, <a href="/search/math?searchtype=author&query=Qian%2C+J">Jianliang 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="2401.01423v2-abstract-short" style="display: inline;"> Starting from the Kirchhoff-Huygens representation and Duhamel's principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation to develop a short-time valid propagato… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.01423v2-abstract-full').style.display = 'inline'; document.getElementById('2401.01423v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.01423v2-abstract-full" style="display: none;"> Starting from the Kirchhoff-Huygens representation and Duhamel's principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation to develop a short-time valid propagator. Second, using the Fourier transform in time, we derive the corresponding Eulerian short-time propagator in frequency domain; on top of this propagator, we further develop a time-frequency-time (TFT) method for the Cauchy problem of time-domain wave equations. Third, we further propose the time-frequency-time-frequency (TFTF) method for the corresponding point-source Helmholtz equation, which provides Green's functions of the Helmholtz equation for all angular frequencies within a given frequency band. Fourth, to implement TFT and TFTF methods efficiently, we introduce butterfly algorithms to compress oscillatory integral kernels at different frequencies. As a result, the proposed methods can construct wave field beyond caustics implicitly and advance spatially overturning waves in time naturally with quasi-optimal computational complexity and memory usage. Furthermore, once constructed the Hadamard integrators can be employed to solve both time-domain wave equations with various initial conditions and frequency-domain wave equations with different point sources. Numerical examples for two-dimensional wave equations illustrate the accuracy and efficiency of the proposed methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.01423v2-abstract-full').style.display = 'none'; document.getElementById('2401.01423v2-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages, 16 figures, 4 tables</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65M80; 65Y20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.16961">arXiv:2312.16961</a> <span> [<a href="https://arxiv.org/pdf/2312.16961">pdf</a>, <a href="https://arxiv.org/ps/2312.16961">ps</a>, <a href="https://arxiv.org/format/2312.16961">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/cnt.2024.13.299">10.2140/cnt.2024.13.299 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On connectedness in the parametric geometry of numbers </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yuming Wei</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Han Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.16961v4-abstract-short" style="display: inline;"> Via multilinear algebra, we formulate a criterion for connectedness in the parametric geometry of numbers in terms of pencils, which are certain algebraic varieties in the space of matrices. As a consequence, we obtain a connectedness result for generic lattices arising from Diophantine approximation on analytic submanifolds, and sharpen Schmidt and Summerer's results of connectedness on simultane… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.16961v4-abstract-full').style.display = 'inline'; document.getElementById('2312.16961v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.16961v4-abstract-full" style="display: none;"> Via multilinear algebra, we formulate a criterion for connectedness in the parametric geometry of numbers in terms of pencils, which are certain algebraic varieties in the space of matrices. As a consequence, we obtain a connectedness result for generic lattices arising from Diophantine approximation on analytic submanifolds, and sharpen Schmidt and Summerer's results of connectedness on simultaneous Diophantine approximation and approximation by linear forms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.16961v4-abstract-full').style.display = 'none'; document.getElementById('2312.16961v4-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> 24 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Incorporate referee's suggestions</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11H06; 11J13; 37B05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Comb. Number Th. 13 (2024) 299-315 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.17507">arXiv:2311.17507</a> <span> [<a href="https://arxiv.org/pdf/2311.17507">pdf</a>, <a href="https://arxiv.org/ps/2311.17507">ps</a>, <a href="https://arxiv.org/format/2311.17507">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"> Computation of outer inverse of tensors based on $t$-product </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Behera%2C+R">Ratikanta Behera</a>, <a href="/search/math?searchtype=author&query=Sahoo%2C+J+K">Jajati Keshari Sahoo</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.17507v1-abstract-short" style="display: inline;"> Tensor operations play an essential role in various fields of science and engineering, including multiway data analysis. In this study, we establish a few basic properties of the range and null space of a tensor using block circulant matrices and the discrete Fourier matrix. We then discuss the outer inverse of tensors based on $t$-product with a prescribed range and kernel of third-order tensors.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.17507v1-abstract-full').style.display = 'inline'; document.getElementById('2311.17507v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.17507v1-abstract-full" style="display: none;"> Tensor operations play an essential role in various fields of science and engineering, including multiway data analysis. In this study, we establish a few basic properties of the range and null space of a tensor using block circulant matrices and the discrete Fourier matrix. We then discuss the outer inverse of tensors based on $t$-product with a prescribed range and kernel of third-order tensors. We address the relation of this outer inverse with other generalized inverses, such as the Moore-Penrose inverse, group inverse, and Drazin inverse. In addition, we present a few algorithms for computing the outer inverses of the tensors. In particular, a $t$-QR decomposition based algorithm is developed for computing the outer inverses. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.17507v1-abstract-full').style.display = 'none'; document.getElementById('2311.17507v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.14766">arXiv:2311.14766</a> <span> [<a href="https://arxiv.org/pdf/2311.14766">pdf</a>, <a href="https://arxiv.org/format/2311.14766">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Reinforcement Learning from Statistical Feedback: the Journey from AB Testing to ANT Testing </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+F">Feiyang Han</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yimin Wei</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Z">Zhaofeng Liu</a>, <a href="/search/math?searchtype=author&query=Qi%2C+Y">Yanxing Qi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.14766v1-abstract-short" style="display: inline;"> Reinforcement Learning from Human Feedback (RLHF) has played a crucial role in the success of large models such as ChatGPT. RLHF is a reinforcement learning framework which combines human feedback to improve learning effectiveness and performance. However, obtaining preferences feedback manually is quite expensive in commercial applications. Some statistical commercial indicators are usually more… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.14766v1-abstract-full').style.display = 'inline'; document.getElementById('2311.14766v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.14766v1-abstract-full" style="display: none;"> Reinforcement Learning from Human Feedback (RLHF) has played a crucial role in the success of large models such as ChatGPT. RLHF is a reinforcement learning framework which combines human feedback to improve learning effectiveness and performance. However, obtaining preferences feedback manually is quite expensive in commercial applications. Some statistical commercial indicators are usually more valuable and always ignored in RLHF. There exists a gap between commercial target and model training. In our research, we will attempt to fill this gap with statistical business feedback instead of human feedback, using AB testing which is a well-established statistical method. Reinforcement Learning from Statistical Feedback (RLSF) based on AB testing is proposed. Statistical inference methods are used to obtain preferences for training the reward network, which fine-tunes the pre-trained model in reinforcement learning framework, achieving greater business value. Furthermore, we extend AB testing with double selections at a single time-point to ANT testing with multiple selections at different feedback time points. Moreover, we design numerical experiences to validate the effectiveness of our algorithm framework. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.14766v1-abstract-full').style.display = 'none'; document.getElementById('2311.14766v1-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> 24 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.10312">arXiv:2311.10312</a> <span> [<a href="https://arxiv.org/pdf/2311.10312">pdf</a>, <a href="https://arxiv.org/ps/2311.10312">ps</a>, <a href="https://arxiv.org/format/2311.10312">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"> Mean Field Games with infinitely degenerate diffusion and non-coercive Hamiltonian </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+Y">Yiming Jiang</a>, <a href="/search/math?searchtype=author&query=Ren%2C+J">Jingchuang Ren</a>, <a href="/search/math?searchtype=author&query=Wei%2C+Y">Yawei Wei</a>, <a href="/search/math?searchtype=author&query=Xue%2C+J">Jie Xue</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.10312v1-abstract-short" style="display: inline;"> In this paper, we consider a class of infinitely degenerate partial differential systems to obtain the Nash equilibria in the mean field games. The degeneracy in the diffusion and the Hamiltonian may be different. This feature brings difficulties to the uniform boundness of the solutions, which is central to the existence and regularity results. First, from the perspective of the value function in… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.10312v1-abstract-full').style.display = 'inline'; document.getElementById('2311.10312v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.10312v1-abstract-full" style="display: none;"> In this paper, we consider a class of infinitely degenerate partial differential systems to obtain the Nash equilibria in the mean field games. The degeneracy in the diffusion and the Hamiltonian may be different. This feature brings difficulties to the uniform boundness of the solutions, which is central to the existence and regularity results. First, from the perspective of the value function in the stochastic optimal control problems, we prove the Lipschitz continuity and the semiconcavity for the solutions of the Hamilton-Jacobi equations (HJE). Then the existence of the weak solutions for the degenerate systems is obtained via a vanishing viscosity method. Furthermore, by constructing an auxiliary function, we conclude the regularity of the viscosity solution for the HJE in the almost everywhere sense. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.10312v1-abstract-full').style.display = 'none'; document.getElementById('2311.10312v1-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 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q89; 35K65; 35A01 </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=Wei%2C+Y&start=50" 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