Infinite-Dimensional Hamilton–Jacobi Equations for Statistical Inference on Sparse Graphs

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-07-01 DOI:10.1137/22m1527696

Tomas Dominguez, Jean-Christophe Mourrat

{"title":"Infinite-Dimensional Hamilton–Jacobi Equations for Statistical Inference on Sparse Graphs","authors":"Tomas Dominguez, Jean-Christophe Mourrat","doi":"10.1137/22m1527696","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4530-4593, August 2024. <br/> Abstract. We study the well-posedness of an infinite-dimensional Hamilton–Jacobi equation posed on the set of nonnegative measures and with a monotonic nonlinearity. Our results will be used in a companion work to propose a conjecture and prove partial results concerning the asymptotic mutual information in the assortative stochastic block model in the sparse regime. The equation we consider is naturally stated in terms of the Gateaux derivative of the solution, unlike previous works in which the derivative is usually of transport type. We introduce an approximating family of finite-dimensional Hamilton–Jacobi equations and use the monotonicity of the nonlinearity to show that no boundary condition needs to be prescribed to establish well-posedness. The solution to the infinite-dimensional Hamilton–Jacobi equation is then defined as the limit of these approximating solutions. In the special setting of a convex nonlinearity, we also provide a Hopf–Lax variational representation of the solution.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Mathematical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m1527696","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4530-4593, August 2024.
Abstract. We study the well-posedness of an infinite-dimensional Hamilton–Jacobi equation posed on the set of nonnegative measures and with a monotonic nonlinearity. Our results will be used in a companion work to propose a conjecture and prove partial results concerning the asymptotic mutual information in the assortative stochastic block model in the sparse regime. The equation we consider is naturally stated in terms of the Gateaux derivative of the solution, unlike previous works in which the derivative is usually of transport type. We introduce an approximating family of finite-dimensional Hamilton–Jacobi equations and use the monotonicity of the nonlinearity to show that no boundary condition needs to be prescribed to establish well-posedness. The solution to the infinite-dimensional Hamilton–Jacobi equation is then defined as the limit of these approximating solutions. In the special setting of a convex nonlinearity, we also provide a Hopf–Lax variational representation of the solution.

查看原文本刊更多论文

用于稀疏图上统计推断的无穷维汉密尔顿-雅可比方程

SIAM 数学分析期刊》，第 56 卷第 4 期，第 4530-4593 页，2024 年 8 月。摘要。我们研究了在非负度量集合上提出的、具有单调非线性的无穷维汉密尔顿-贾可比方程的好求性。我们的研究结果将被用于另一项研究，以提出一个猜想，并证明有关稀疏状态下同类随机块模型中渐近互信息的部分结果。我们所考虑的方程是以解的盖陶导数来自然表述的，这与以前的工作不同，在以前的工作中，导数通常是传输类型的。我们引入了有限维汉密尔顿-雅可比方程的近似族，并利用非线性的单调性证明无需规定边界条件即可建立良好求解。然后，无限维 Hamilton-Jacobi 方程的解被定义为这些近似解的极限。在凸非线性的特殊情况下，我们还提供了解的霍普夫-拉克斯变分表示。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

SIAM Journal on Mathematical Analysis 数学-应用数学

CiteScore

3.30

自引率

5.00%

发文量

175

审稿时长

12 months

期刊介绍： SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.