arXiv research

The paper develops inequalities for log-concave functions and related surface areas.

We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set f…

We explain a general construction through which concave elliptic operators on complex manifolds give rise to concave functions on cohomology. In particular, this leads to generalized versions of the Khovanskii-Teissier inequalities.

Establishes a concavity property for positive Hessian quotient operators.

The paper proves extension theorems for complex manifolds with Levi $q$-concave domains.

Derives concavity inequality and estimates for $k$-Hessian equations.

In this paper, we prove the concavity of $p$-entropy power of probability densities solving the $p$-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of $L^p$-Euclidean Nash inequality and $L^p$-Euclidean Logarithmic Sobolev inequality, moreover, an improv…

Minimal graph level sets are concave if boundary is concave.

We prove the concavity of $p$-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on $\mathbb{R}^n$ or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp $L^p$-Sobolev inequality and $L^p$-Gagliardo-Nirenberg inequalities on…

Paper proves generalized Talagrand inequality for Sinkhorn distance.

Improved algorithms for convex-concave min-max optimization and monotone variational inequalities.

Estimates for complex Hessian equations on Hermitian manifolds.

Paper estimates curvature of semi-convex solutions in hyperbolic space.

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

New Orlicz Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are…

Two important goals of high-dimensional modeling are prediction and variable selection. In this article, we consider regularization with combined $L_1$ and concave penalties, and study the sampling properties of the global optimum of the suggested method in ultra-high dimensional settings. The $L_1$-penalty provides th…

We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operato…

New inequalities generalize Li's theorem on mixed Hodge structures.

Study on a metric space derived from Kähler manifolds.

The paper proves a Jensen's inequality in spaces with lower bounded curvature.

Information concentration of probability measures have important implications in learning theory. Recently, it is discovered that the information content of a log-concave distribution concentrates around their differential entropy, albeit with an unpleasant dependence on the ambient dimension. In this work, we prove th…

Improved sampling algorithm with state-of-the-art complexity bounds.

By using asymptotic Morse inequalities we give a lower bound for the space of holomorphic sections of high tensor powers in a positive line bundle over a q-concave domain. The curvature of the positive bundle induces a hermitian metric on the manifold. The bound is given explicitely in terms of the volume of the domain…

Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.

High-dimensional data analysis has motivated a spectrum of regularization methods for variable selection and sparse modeling, with two popular classes of convex ones and concave ones. A long debate has been on whether one class dominates the other, an important question both in theory and to practitioners. In this pape…

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

New sampling algorithms for complex distributions without log-concavity.

We consider variational inequalities coming from monotone operators, a setting that includes convex minimization and convex-concave saddle-point problems. We assume an access to potentially noisy unbiased values of the monotone operators and assess convergence through a compatible gap function which corresponds to the …

The paper establishes inequalities for Chern classes and Riemann-Roch type inequalities for projective manifolds.

We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in the high tensor powers of the line bund…

In this paper, we present a simple analysis of {\bf fast rates} with {\it high probability} of {\bf empirical minimization} for {\it stochastic composite optimization} over a finite-dimensional bounded convex set with exponential concave loss functions and an arbitrary convex regularization. To the best of our knowledg…

In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this paper, the Jacobian determinant of a gradient map is shown to be log-concave with respect to a convex combination of the potential functions when the underlying manifold is t…

McDiarmid's inequality under dependence via approximate tensorization of entropy

New method improves sampling for weakly log-concave posteriors.

The paper finds the unique minimizer of area for hyperbolic bodies with curvature constraints.

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

The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.

Study optimal consumption for loss-averse agents considering past spending peaks.

Unified study of Brunn-Minkowski conjectures for log-concave measures.

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $γ$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex…

New transport method simplifies cutoff phenomenon for Markov processes.

Universal tester-learner for halfspaces over structured distributions.

Algorithm samples from composite log-concave distributions using gradient evaluations and restricted Gaussian oracles.

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

New algorithms reduce variance in solving complex mathematical problems.

The paper proves optimizability implies inequalities for sampling.

ULA estimates covariance of log-concave distributions efficiently.

New sampling method guarantees approximate first-order stationary points for non-convex functions.