arXiv research

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

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

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

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…

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

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…

LMC algorithm receives first convergence guarantees under weak smoothness conditions.

The paper derives inequalities and formulas for generalized Ricci flow.

New algorithm samples superlinearly growing log-gradient distributions.

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

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

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

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

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

The study proves inequalities and curvature properties for Markov chains.

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…

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 …

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

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

Paper disproves conjecture about log-Sobolev constants.

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

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we ch…

Greedy MI maximization method outperforms existing approaches in nonlinear models.

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…

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…

The paper proves optimizability implies inequalities for sampling.

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

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

Sharp bounds on uniform generalization errors in binary linear classification.

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

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…

Our work improves Langevin dynamics convergence on manifolds.

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

The paper derives Harnack inequalities for evolving Riemannian manifolds without dimensionality restrictions.

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.

Derives sub-Riemannian Ricci curvature for various manifolds.

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

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

We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a …

Sharp inequality found on three-balls for fourth order Sobolev traces.

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…

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.

New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.

Parallel sampling for smooth distributions with fast convergence.

Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.

Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.