arXiv research

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

The study proves inequalities and curvature properties for Markov chains.

LMC algorithm receives first convergence guarantees under weak smoothness conditions.

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

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

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

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

The paper derives inequalities and formulas for generalized Ricci flow.

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…

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…

Unified analysis of MPLE for Ising models with bounded operator norm or infinity norm.

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

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

New sampling and identity-testing methods for mixtures of distributions that don't satisfy approximate tensorization of entropy.

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

New algorithm samples superlinearly growing log-gradient distributions.

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

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

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…

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 …

Improved Langevin algorithms with prior diffusion achieve dimension-independent convergence for non-log-concave distributions.

New sampling algorithms for complex distributions without log-concavity.

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

Paper disproves conjecture about log-Sobolev constants.

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…

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…

Sharp bounds on uniform generalization errors in binary linear classification.

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…

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

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

Our work improves Langevin dynamics convergence on manifolds.

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

Greedy MI maximization method outperforms existing approaches in nonlinear models.

The paper proves optimizability implies inequalities for sampling.

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

Derives sub-Riemannian Ricci curvature for various manifolds.

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 …

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

Improved Langevin Monte Carlo reduces energy barriers for faster optimization.

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

Extends Langevin dynamics for constrained domains.

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…

Mirror Langevin Algorithm converges with zero bias.

Improved sampling from mean-field stationary distributions.

We study Bakry-Emery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the $Γ_2$ operator may not be bounded from below but the horizontal Bakry-Emery curvature is. As we prove it, under a bracket generating condition, this weaker …

The Riemannian Langevin Algorithm samples from manifolds efficiently.