arXiv research

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

Ancient Ricci flows are identified without curvature sign condition.

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…

The paper derives inequalities and formulas for generalized Ricci flow.

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…

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

In this paper we introduce the log entropy functional and establish its monotonicity along the Ricci flow. One consequence of it is the monotonicity of the logarithmic Sobolev constant along the Ricci flow.

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

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

Our work improves Langevin dynamics convergence on manifolds.

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

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

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…

Paper disproves conjecture about log-Sobolev constants.

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

LMC algorithm receives first convergence guarantees under weak smoothness conditions.

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…

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

New algorithm samples superlinearly growing log-gradient distributions.

Adapts to estimate functions from noisy ERT data.

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

The study proves inequalities and curvature properties for Markov chains.

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

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…

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…

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

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 …

Study fine Pólya-Szegő inequalities in metric spaces with applications.

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

Proposes a new generalization bound for Bayesian deep nets without strict assumptions.

Improved sampling from mean-field stationary distributions.

Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $Ω$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geome…

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

Improved Langevin Monte Carlo reduces energy barriers for faster optimization.

Extends Langevin dynamics for constrained domains.

Greedy MI maximization method outperforms existing approaches in nonlinear models.

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

The paper proves optimizability implies inequalities for sampling.

By using Hsu's multiplicative functional for the Neumann heat equation, a natural damped gradient operator is defined for the reflecting Brownian motion on compact manifolds with boundary. This operator is linked to quasi-invariant flows in terms of a integration by parts formula, which leads to the standard log-Sobole…

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 $-…

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…

New method samples from non-log-concave distributions with weak dissipativity.

For sub-Riemannian manifolds with a chosen complement, we first establish the derivative formula and integration by parts formula on path space with respect to a natural gradient operator. By using these formulae, we then show that upper and lower bounds of the horizontal Ricci curvature correspond to functional inequa…

New sampling algorithms for complex distributions without log-concavity.

Score matching efficiency tied to distribution isoperimetric properties.

Study continuity of complex Sobolev functions, with applications to Kaehler metrics.

The paper analyzes convergence rates of Langevin dynamics and Proximal Sampler using $Φ$-divergence.

Researchers prove rigidity for log-Sobolev inequality on specific metric spaces.