arXiv research

Paper disproves conjecture about log-Sobolev constants.

Sharp log-Sobolev inequalities proved for ${\sf CD}(0,N)$ spaces.

Let $P_t$ be the diffusion semigroup generated by $L:=Δ+\nabla V$ on a complete connected Riemannian manifold with $\operatorname {Ric}\ge-(σ^2ρ_o^2+c)$ for some constants $σ, c>0$ and $ρ_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the…

New algorithm samples superlinearly growing log-gradient distributions.

We derive weighted log-Sobolev inequalities from a class of super Poincaré inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\d…

The study proves inequalities and curvature properties for Markov chains.

Improved Langevin Monte Carlo reduces energy barriers for faster optimization.

The paper extends log-Sobolev inequalities to matrix-valued settings using combinatorial methods.

Log-Sobolev inequality proven for submanifolds in specific types of manifolds.

Sharp bounds on uniform generalization errors in binary linear classification.

The paper proves optimizability implies inequalities for sampling.

Sharp Lipschitz bounds for flow-matching and diffusion models with optimal sampling rates.

Score matching efficiency tied to distribution isoperimetric properties.

Improved sampling from non-log-concave distributions with polynomial query complexity.

In this note we prove a new ε-regularity theorem for the Ricci flow. Let (M^n,g(t)) with t\in [-T,0] be a Ricci flow and H_{x} the conjugate heat kernel centered at a point (x,0) in the final time slice. Substituting H_{x} into Perelman's W-functional produces a monotone function W_{x}(s) of s \in [-T,0], the pointed e…

Polynomial convergence proved for SGM, improving over previous methods.

LMC algorithm receives first convergence guarantees under weak smoothness conditions.

Study establishes Pólya-Szegő inequalities on submanifolds with small total mean curvature.

New deficit functions link elliptic and parabolic inequalities, proving log Sobolev.

In this paper, we study the gradient estimate for positive solutions to the following nonlinear heat equation problem $u_t-Δu=au\log u+Vu, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative Ricci curvature. Here $a\leq 0$ is a constant, $V$ is a smooth function on $M$ with $-…

Study non-negative curvature Markov chains, proving entropy contraction.

The paper derives inequalities and formulas for generalized Ricci flow.

Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.

New framework improves EM algorithm convergence under log-Sobolev inequality.

In this paper, we study elliptic gradient estimates for a nonlinear $f$-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton's and Souplet-Zhang's gradient estimates for positive solutions to the nonlinear $f$-…

We develop a Malliavin calculus on the horizontal path space of a totally geodesic Riemannian foliation. As a first application, under suitable assumptions, we prove a log-Sobolev inequality for a natural one-parameter family of infinite-dimensional Ornstein-Uhlenbeck type operators. As a second application, we obtain …

Ancient Ricci flows are identified without curvature sign condition.

Let $\M$ be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (\ref{logfanhan}) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This…

In this paper, we prove the equivalent of ultracontractive bound of heat semigroup or the uniform upper bound of the heat kernel with the Nash inequality, Log-Sobolev inequalities on graphs. We also show that under the assumption of volume growth and nonnegative curvature $CDE'(n,0)$ the Sobolev inequality, Nash inequa…

Proves error bounds for PGD, extending log-Sobolev and Talagrand inequalities.

We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution $ν= e^{-f}$ on $\mathbb{R}^n$ under isoperimetry. We prove a convergence guarantee for PLA in Kullback-Leibler (KL) divergence when $ν$ satisfies log-Sobolev inequality (LSI) and $f$ has bounded second and third derivatives. Thi…

Our work improves Langevin dynamics convergence on manifolds.

New method controls classifier guidance in diffusion models.

The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.

In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem $u_t-Δu=au\log u, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative (Bakry-Emery)-Ricci curvature. Here…

We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution $ν= e^{-f}$ on $\mathbb{R}^n$. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming $ν$ satisfies a log-Sobolev inequality and the Hessian of $f$ is bounded. Notably, we do not assume convexity or boun…

This paper proves a Liouville type result for a specific higher-order equation on the sphere.

Paper proposes new Langevin samplers for sampling from log-concave distributions with superlinear gradient growth.

Sharp Gaussian isoperimetry proven along Ricci flow.

Derives sub-Riemannian Ricci curvature for various manifolds.

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

Inequalities linking entropy, Fisher info, Stein discrepancy, and Wasserstein distance on Riemannian manifolds.

Improved sampling guarantees for underdamped Langevin Monte Carlo without restrictive assumptions.

Given a probability measure $μ$ supported on a convex subset $Ω$ of Euclidean space $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincaré and log-Sobolev type inequalities on $(Ω,g_0,μ)$. To this end, we change the metric $g_0$ to a more general Riemannian one $g$, adapted in a certain sense to $μ$, and perform…

Greedy MI maximization method outperforms existing approaches in nonlinear models.

In this paper we establish the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity. When Ricci curvature is also bounded from above we get exponential decay at infinity o…

Extends Langevin dynamics for constrained domains.

New bound relaxes uniform gradient norm assumptions for PAC-Bayesian bounds.