Communications on Pure and Applied Mathematics最新文献_第2页

On learning Gaussian multi-index models with gradient flow part I: General properties and two-timescale learning 关于梯度流高斯多指标模型的学习，第一部分：一般性质和双时间尺度学习

IF 2.7 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2025-07-15 DOI: 10.1002/cpa.70006

Alberto Bietti, Joan Bruna, Loucas Pillaud-Vivien

引用次数: 0

Quasi-invariance of Gaussian measures for the 3 d $3d$ energy critical nonlinear Schrödinger equation 三维$三维$能量临界非线性Schrödinger方程高斯测度的拟不变性

IF 2.7 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2025-06-26 DOI: 10.1002/cpa.70001

Chenmin Sun, Nikolay Tzvetkov

{"title":"Quasi-invariance of Gaussian measures for the \u0000 \u0000 \u0000 3\u0000 d\u0000 \u0000 $3d$\u0000 energy critical nonlinear Schrödinger equation","authors":"Chenmin Sun, Nikolay Tzvetkov","doi":"10.1002/cpa.70001","DOIUrl":"10.1002/cpa.70001","url":null,"abstract":"We consider the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$3d$</annotation>\u0000 </semantics></math> energy critical nonlinear Schrödinger equation with data distributed according to the Gaussian measure with covariance operator <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>−</mo>\u0000 <mi>Δ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mi>s</mi>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$(1-Delta)^{-s}$</annotation>\u0000 </semantics></math>, where <math>\u0000 <semantics>\u0000 <mi>Δ</mi>\u0000 <annotation>$Delta$</annotation>\u0000 </semantics></math> is the Laplace operator and <math>\u0000 <semantics>\u0000 <mi>s</mi>\u0000 <annotation>$s$</annotation>\u0000 </semantics></math> is sufficiently large. We prove that the flow sends full measure sets to full measure sets. We also discuss some simple applications. This extends a previous result by Planchon-Visciglia and the second author from <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$1d$</annotation>\u0000 </semantics></math> to higher dimensions.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 12","pages":"2305-2353"},"PeriodicalIF":2.7,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.70001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144500756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Bogomolov–Gieseker inequality for log terminal Kähler threefolds 日志终端Kähler的三倍Bogomolov-Gieseker不等式

IF 2.7 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2025-06-23 DOI: 10.1002/cpa.70000

Henri Guenancia, Mihai Păun

引用次数: 0

Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUE 有条件GinUE的自由能展开式和最小特征值的大偏差

IF 2.7 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2025-06-23 DOI: 10.1002/cpa.70005

Sung-Soo Byun, Seong-Mi Seo, Meng Yang

{"title":"Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUE","authors":"Sung-Soo Byun, Seong-Mi Seo, Meng Yang","doi":"10.1002/cpa.70005","DOIUrl":"10.1002/cpa.70005","url":null,"abstract":"We consider a planar Coulomb gas ensemble of size <math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> with the inverse temperature <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>β</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$beta =2$</annotation>\u0000 </semantics></math> and external potential <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Q</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>z</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>z</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 <mi>c</mi>\u0000 <mi>log</mi>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>z</mi>\u0000 <mo>−</mo>\u0000 <mi>a</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$Q(z)=|z|^2-2c log |z-a|$</annotation>\u0000 </semantics></math>, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>c</mi>\u0000 <mo>></mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$c>0$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>∈</mo>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation>$a in mathbb {C}$</annotation>\u0000 </semantics></math>. Equivalently, this model can be realised as <math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> eigenvalues of the complex Ginibre matrix of size <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>c</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 <mi>N</mi>\u0000 <mo>×</mo>\u0000 <mo>(</mo>\u0000 <mi>c</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$(c+1) N times","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 12","pages":"2247-2304"},"PeriodicalIF":2.7,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.70005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144370462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The Brownian loop measure on Riemann surfaces and applications to length spectra 黎曼曲面上的布朗环测量及其在长度谱上的应用

IF 2.7 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2025-06-20 DOI: 10.1002/cpa.70003

Yilin Wang, Yuhao Xue

引用次数: 0

Understanding the training of infinitely deep and wide ResNets with conditional optimal transport 了解具有条件最优传输的无限深和无限宽ResNets的训练

IF 2.7 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2025-06-20 DOI: 10.1002/cpa.70004

Raphaël Barboni, Gabriel Peyré, François-Xavier Vialard

{"title":"Understanding the training of infinitely deep and wide ResNets with conditional optimal transport","authors":"Raphaël Barboni, Gabriel Peyré, François-Xavier Vialard","doi":"10.1002/cpa.70004","DOIUrl":"10.1002/cpa.70004","url":null,"abstract":"We study the convergence of gradient flow for the training of deep neural networks. While residual neural networks (ResNet) are a popular example of very deep architectures, their training constitutes a challenging optimization problem, notably due to the non-convexity and the non-coercivity of the objective. Yet, in applications, such tasks are successfully solved by simple optimization algorithms such as gradient descent. To better understand this phenomenon, we focus here on a “mean-field” model of an infinitely deep and arbitrarily wide ResNet, parameterized by probability measures on the product set of layers and parameters, and with constant marginal on the set of layers. Indeed, in the case of shallow neural networks, mean field models have been proven to benefit from simplified loss landscapes and good theoretical guarantees when trained with gradient flow w.r.t. the Wasserstein metric on the set of probability measures. Motivated by this approach, we propose to train our model with gradient flow w.r.t. the conditional optimal transport (COT) distance: a restriction of the classical Wasserstein distance which enforces our marginal condition. Relying on the theory of gradient flows in metric spaces, we first show the well-posedness of the gradient flow equation and its consistency with the training of ResNets at finite width. Performing a local Polyak–Łojasiewicz analysis, we then show convergence of the gradient flow for well-chosen initializations: if the number of features is finite but sufficiently large and the risk is sufficiently small at initialization, the gradient flow converges to a global minimizer. This is the first result of this type for infinitely deep and arbitrarily wide ResNets. In addition, this work is an opportunity to study in more detail the COT metric, particularly its dynamic formulation. Some of our results in this direction might be interesting on their own.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 11","pages":"2149-2205"},"PeriodicalIF":2.7,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144334962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Boundary conditions and universal finite‐size scaling for the hierarchical |φ|4$|varphi |^4$ model in dimensions 4 and higher 4维及以上层次|φ|4$|varphi |^4$模型的边界条件和通用有限尺寸标度

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2025-04-29 DOI: 10.1002/cpa.22256

Emmanuel Michta, Jiwoon Park, Gordon Slade

{"title":"Boundary conditions and universal finite‐size scaling for the hierarchical |φ|4$|varphi |^4$ model in dimensions 4 and higher","authors":"Emmanuel Michta, Jiwoon Park, Gordon Slade","doi":"10.1002/cpa.22256","DOIUrl":"https://doi.org/10.1002/cpa.22256","url":null,"abstract":"We analyse and clarify the finite‐size scaling of the weakly‐coupled hierarchical ‐component model for all integers in all dimensions , for both free and periodic boundary conditions. For , we prove that for a volume of size with periodic boundary conditions the infinite‐volume critical point is an effective finite‐volume critical point, whereas for free boundary conditions the effective critical point is shifted smaller by an amount of order . For both boundary conditions, the average field has the same non‐Gaussian limit within a critical window of width around the effective critical point, and in that window we compute the universal scaling profile for the susceptibility. In contrast, and again for both boundary conditions, the average field has a massive Gaussian limit when above the effective critical point by an amount . In particular, at the infinite‐volume critical point the susceptibility scales as for periodic boundary conditions and as for free boundary conditions. We identify a mass generation mechanism for free boundary conditions that is responsible for this distinction and which we believe has wider validity, in particular to Euclidean (non‐hierarchical) models on in dimensions . For we prove a similar picture with logarithmic corrections. Our analysis is based on the rigorous renormalisation group method of Bauerschmidt, Brydges and Slade, which we improve and extend.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"88 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143889831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Persistence and ball exponents for Gaussian stationary processes 高斯平稳过程的持续性和球指数

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2025-04-29 DOI: 10.1002/cpa.22255

Naomi D. Feldheim, Ohad N. Feldheim, Sumit Mukherjee

引用次数: 0

Boundary statistics for the six‐vertex model with DWBC 带有DWBC的六顶点模型的边界统计

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2025-04-24 DOI: 10.1002/cpa.22254

Vadim Gorin, Karl Liechty

引用次数: 0

On classification of global dynamics for energy-critical equivariant harmonic map heat flows and radial nonlinear heat equation 能量临界等变调和映射热流和径向非线性热方程的全局动力学分类

IF 3.1 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2025-04-18 DOI: 10.1002/cpa.22253

Kihyun Kim, Frank Merle

{"title":"On classification of global dynamics for energy-critical equivariant harmonic map heat flows and radial nonlinear heat equation","authors":"Kihyun Kim, Frank Merle","doi":"10.1002/cpa.22253","DOIUrl":"10.1002/cpa.22253","url":null,"abstract":"We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation (NLH). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 <mo>≥</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$Dge 3$</annotation>\u0000 </semantics></math>; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of <math>\u0000 <semantics>\u0000 <msup>\u0000 <mover>\u0000 <mi>H</mi>\u0000 <mo>̇</mo>\u0000 </mover>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$dot{H}^{1}$</annotation>\u0000 </semantics></math>-bounded radial solutions to (NLH) in dimensions <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>≥</mo>\u0000 <mn>7</mn>\u0000 </mrow>\u0000 <annotation>$Nge 7$</annotation>\u0000 </semantics></math>, building upon soliton resolution for such solutions. To our knowledge, this provides the first rigorous classification of bubble tree dynamics within symmetry. We introduce a new approach based on the energy method that does not rely on maximum principle. The key ingredient of the proof is a monotonicity estimate near any bubble tree configurations, which in turn requires a delicate construction of modified multi-bubble profiles also.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 9","pages":"1783-1842"},"PeriodicalIF":3.1,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22253","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143847057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0