Related papers: Log-concavity and strong log-concavity: a review
In this paper we study the joint convexity/concavity of the trace functions \[ \Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, \] where $A$ and $B$ are positive definite matrices and $K$ is any…
In a recent work, Bo'az Klartag showed that, given a convex body with minimal volume product, its isotropic constant is related to its volume product. As a consequence, he obtained that a strong version of the slicing conjecture implies…
This note contributes to the understanding of generalized entropy power inequalities. Our main goal is to construct a counter-example regarding monotonicity and entropy comparison of weighted sums of independent identically distributed…
We study the linear convergence of variants of the Frank-Wolfe algorithms for some classes of strongly convex problems, using only affine-invariant quantities. As in Guelat & Marcotte (1986), we show the linear convergence of the standard…
We prove log-concavity for the function counting partitions without sequences. We use an exact formula for a mixed-mock modular form of weight zero, explicit estimates on modified Kloosterman sums and analytic techniques. Finally, we…
We show that recent multivariate generalizations of the Araki-Lieb-Thirring inequality and the Golden-Thompson inequality [Sutter, Berta, and Tomamichel, Comm. Math. Phys. (2016)] for Schatten norms hold more generally for all unitarily…
Our aim is to provide a short and self contained synthesis which generalise and unify various related and unrelated works involving what we call Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies can be seen in…
In this paper, we prove coincidence and common fixed points results under nonlinear contractions on a metric space equipped with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially…
In this paper, we review recent results on stability and instability in logarithmic Sobolev inequalities, with a particular emphasis on strong norms. We consider several versions of these inequalities on the Euclidean space, for the…
Bernstein-von Mises results (BvM) establish that the Laplace approximation is asymptotically correct in the large-data limit. However, these results are inappropriate for computational purposes since they only hold over most, and not all,…
We give a new proof of the sharp form of Young's inequality for convolutions, first proved by Beckner [Be] and Brascamp-Lieb [BL]. The latter also proved a sharp reverse inequality in the case of exponents less than $1$. Our proof is…
We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $\Omega \subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial \Omega$. This reduces the study of the Neumann…
It is shown that for a given log-concave function, its symmetric decreasing rearrangement is always harder to approximate in the symmetric difference metric by inner log-linearizations with a fixed number of break points. This extends a…
An "entropy increasing to the maximum" result analogous to the entropic central limit theorem (Barron 1986; Artstein et al. 2004) is obtained in the discrete setting. This involves the thinning operation and a Poisson limit. Monotonic…
A sequence $\{z_n\}_{n\geq0}$ is called ratio log-convex in the sense that the ratio sequence $\{\frac{z_{n+1}}{z_n}\}_{n\geq0}$ is log-convex. Based on a three-term recurrence for sequences, we develop techniques for dealing with the ratio…
It is well known that if a random vector satisfies a log-Sobolev inequality, all of its marginals have subgaussian tails. In the spirit of the KLS conjecture, we investigate whether this implication can be reversed under a log-concavity…
Mean field variational inference (VI) is the problem of finding the closest product (factorized) measure, in the sense of relative entropy, to a given high-dimensional probability measure $\rho$. The well known Coordinate Ascent Variational…
Motivated, in part, by the desire to develop an information-theoretic foundation for compound Poisson approximation limit theorems (analogous to the corresponding developments for the central limit theorem and for simple Poisson…
In 1970, Donald Ornstein proved a landmark result in dynamical systems, viz., two Bernoulli systems with the same entropy are isomorphic except for a measure 0 set. Keane and Smorodinsky gave a finitary proof of this result. They also…
We consider the class univalent log-harmonic mappings on the unit disk. Firstly, we obtain necessary and sufficient conditions for a complex-valued continuous function to be starlike or convex in the unit disk. Then we present a general…