Logarithmic Sobolev Inequalities

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Sharp Convergence of Wasserstein Gradient Flows for Spectrally Nonnegative Interaction Energies
Sep 22, 2026Zhengjiang Lin, Philippe RigolletWasserstein Gradient Flows Logarithmic Sobolev Inequalities
Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networks
Sep 17, 2026Xianjun Li, Yunfei YangLogarithmic Sobolev Inequalities Rectified Linear Unit Networks
School of Statistics and Data Science, Capital University of Economics and Business, Beijing, China. · School of Mathematics (Zhuhai) and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Zhuhai, China.
Near-Optimal Learning of Gaussian Sobolev Operators
Jul 8, 2026Ben Adcock, Michael Griebel, Gregor MaierProbabilistic Operator Learning Logarithmic Sobolev Inequalities
Department of Mathematics, Simon Fraser University, Burnaby BC, Canada · Institute for Numerical Simulation, University of Bonn, Bonn, Germany · Fraunhofer Institute for Algorithms and Scientific Computing (SCAI), Sankt Augustin, Germany
Tightening the Score Matching Gap for Diffusion Models
Jul 5, 2026Benjamin Dupuis, Tyler Farghly, Maxime Haddouche +2Score-Based Diffusion Model Kullback-Leibler Divergence
INRIA - CNRS - D´epartement d’Informatique de l’Ecole Normale Sup´erieure - PSL Research University, France · Ecole Polytechnique - CMAP, IP Paris, Palaiseau, France
On the stability of scale-space metrics
Jun 25, 2026William LeebWasserstein Distance Homography
School of Mathematics University of Minnesota, Twin Cities Minneapolis, MN
Generalization Guarantees for Multi-Input Neural Operator Learning in Sobolev Spaces
Jun 16, 2026Yahong Yang, Zecheng Zhang, Wei Zhu +2Probabilistic Operator Learning Logarithmic Sobolev Inequalities
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA. · Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA. · Department of Mathematics, Hong Kong Baptist University, FSC1202, Fong Shu Chuen Building, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
Sobolev Approximation by Fixed-Size Neural Networks with Arbitrary Accuracy
Jun 15, 2026Baicheng Li, Haizhao Yang, Shijun ZhangNeural Approximations Logarithmic Sobolev Inequalities
Department of Mathematics, University of Maryland, College Park, MD, USA · Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China
Brownian Kernel Ladders
Jun 14, 2026Mahdi Mohammadigohari, Giuseppe Di Fatta, Giuseppe Nicosia +1Kernel Hilbert Spaces Logarithmic Sobolev Inequalities
Faculty of Engineering, Free University of Bozen-Bolzano, Via Bruno Buozzi 1, Bolzano, 39100, Italy · Department of Biomedical and Biotechnological Sciences, University of Catania, Catania, 95123, Santa Sofia, 97, Italy · Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, FL 32611, Gainesville, USA
Approximation and learning of anisotropic and mixed smooth functions by deep ReLU neural networks
May 29, 2026Yunfei Yang, Jun FanLogarithmic Sobolev Inequalities Rectified Linear Unit Networks
School of Mathematics (Zhuhai) and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Zhuhai, China. · Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, China.
Random Neural Network Expressivity for Non-Linear Partial Differential Equations
May 24, 2026Muhammed Ali Mehmood, Lukas GononPartial Differential Equations Neural Approximations
Department of Mathematics Imperial College London UK · School of Computer Science University of St. Gallen Switzerland
Coloring the Noise: Adversarial Sobolev Alignment for Faithful Image Super Resolution
May 22, 2026Hongbo Wang, Huaibo Huang, Pin Wang +3Image Restoration Logarithmic Sobolev Inequalities
MAIS & NLPR, Institute of Automation, Chinese Academy of Sciences, Beijing, China · School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing, China · Kuaishou Technology, Beijing, China
Uniform-in-Time Weak Propagation-of-Chaos in Shallow Neural Networks
May 21, 2026Margalit Glasgow, Joan BrunaTwo-Layer Neural Networks Wasserstein Gradient Flows
Massachusetts Institute of Technology · Courant Institute School of Mathematics, Computing and Data Science, New York University
Posterior Contraction of Lévy Adaptive B-spline Regression in Besov Spaces
May 19, 2026Jeunghun Oh, Sewon Park, Jaeyong LeeSplines Logarithmic Sobolev Inequalities
Department of Statistics, Seoul National University. · Department of Statistics, Sookmyung Women’s University.
Shallow ReLU $^s$ Networks in $L^p$ -Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization
May 18, 2026Weizhao Li, Fanghui Liu, Lei ShiLogarithmic Sobolev Inequalities Rectified Linear Unit Networks
School of Mathematical Sciences Fudan University Shanghai, China · School of Mathematical Sciences Institute of Natural Sciences and MOE-LSC2026 Shanghai Jiao Tong University Shanghai, China · School of Mathematical Sciences Shanghai Key Laboratory for Contemporary Applied Mathematics25 Fudan University Shanghai, China
Wasserstein bounds for denoising diffusion probabilistic models via the Föllmer process
May 18, 2026Yuta KoikeWasserstein Distance Denoising Diffusion Probabilistic Model
*Graduate School of Mathematical Sciences, University of Tokyo · †CREST, Japan Science and Technology Agency
A proximal gradient algorithm for composite log-concave sampling
May 12, 2026Linghai Liu, Sinho ChewiLog-Concave Distributions Proximal Operators
Yale University
Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces
May 12, 2026Jeunghun Oh, Kyeongwon Lee, Jaeyong Lee +1Kolmogorov-Arnold Networks Logarithmic Sobolev Inequalities
Department of Statistics, Seoul National University · Department of Statistics, Inha University · Department of Mathematics, University of Maryland
Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems
May 5, 2026Hanfei Zhou, Lei ShiLogarithmic Sobolev Inequalities Neural Approximations
School of Mathematical Sciences, Fudan University, Shanghai, 200433, China · Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China
Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks
May 5, 2026Diego MarcondesPartial Differential Equations Logarithmic Sobolev Inequalities
Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation
May 4, 2026Nicole HaoFourier Neural Operators Logarithmic Sobolev Inequalities
Department of Mathematics, Cornell University
Structured Approximations of Measures
Oct 13, 2023Keaton Hamm, Varun KhuranaProbability Measures Wasserstein Distance
DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF TEXAS AT ARLINGTON, ARLINGTON, TX · DIVISION OF DATA SCIENCE, THE UNIVERSITY OF TEXAS AT ARLINGTON, ARLINGTON, TX · DEPARTMENT OF APPLIED MATHEMATICS, BROWN UNIVERSITY, PROVIDENCE, RI 02906.

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Sharp Convergence of Wasserstein Gradient Flows for Spectrally Nonnegative Interaction Energies

Zhengjiang Lin, Philippe Rigollet

Wasserstein Gradient Flows Logarithmic Sobolev Inequalities Convergence Kernel Method Long-Term

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We study the long-time behavior of Wasserstein gradient flows for interaction energies

\mathsf E[μ] = \frac12\iint_{M\times M}K(x,y)\,\mathrm dμ(x)\,\mathrm dμ(y)

on a closed manifold

M

. For kernels diagonal in a Laplace eigenbasis with nonnegative spectral coefficients, we prove a differential inequality relating the relative entropy to the energy gap. Consequently, for any nonnegative initial density

u_0\in L^p(M)

p>1

, the energy gap is integrable in time and satisfies

\mathsf E[μ_t]-\mathsf E_{\min}=o(t^{-1}).

If all spectral coefficients are positive, the flow converges weakly to the constant measure. These interaction energies need not be geodesically convex in Wasserstein space, and the associated flows contain no diffusion; their global convergence therefore does not follow from standard Wasserstein gradient flow theory. The kernels covered by our results include zonal kernels on spheres, kernels arising in transformer models, regularized Riesz kernels, and inverse fractional Laplacian kernels. We also investigate the sharpness of the

o(t^{-1})

rate. For any smooth kernel in this class with infinitely many positive spectral coefficients and any

δ>0

, we construct a solution of the linearized flow whose energy is comparable to

t^{-1-δ}

along a sequence of times tending to infinity. Moreover, for any

δ>0

, by choosing a suitable inverse fractional Laplacian kernel on the flat torus, we construct an exact solution of the nonlinear Wasserstein gradient flow whose energy is comparable to

t^{-1-δ}

. The nonlinear construction is based on uniform-in-time estimates for the evolution of the dyadic Fourier coefficient blocks and a blockwise energy-persistence argument. These estimates also yield a uniform-in-time quantitative comparison between the nonlinear Wasserstein gradient flow and its linearization.

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