Related papers: Total variation distance estimates via $L^2$-norm …
Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss…
This paper provides estimates for the convergence rate of the total variation distance in the framework of the Breuer-Major theorem, assuming some smoothness properties of the underlying function. The results are proved by applying new…
Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the problem of computing the TV distance of two product distributions over the domain…
Given two high-dimensional Gaussians with the same mean, we prove a lower and an upper bound for their total variation distance, which are within a constant factor of one another.
Mixtures of high dimensional Gaussian distributions have been studied extensively in statistics and learning theory. While the total variation distance appears naturally in the sample complexity of distribution learning, it is analytically…
We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at…
We examine the integrated squared difference, also known as the L2 distance (L2D), between two probability densities. Such a distance metric allows for comparison of differences between pairs of distributions or changes in a distribution…
We review some simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these…
We show that a measure on the real line that is the image of a log-concave measure under a polynomial of degree $d$ possesses a density from the Nikol'skii--Besov class of fractional order $1/d$. This result is used to prove an estimate of…
We propose a method for estimating a log-concave density on $\mathbb R^d$ from samples, under the assumption that there exists an orthogonal transformation that makes the components of the random vector independent. While log-concave…
We consider the problem of computing the distance between two piecewise-linear bivariate functions $f$ and $g$ defined over a common domain $M$. We focus on the distance induced by the $L_2$-norm, that is $\|f-g\|_2=\sqrt{\iint_M (f-g)^2}$.…
The quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories. Motivated by the categorical interpretation of partial metric spaces, we are led to investigate the…
In this paper, we provide explicit lower bounds with respect to some quantities of interest (parameters of the underlying distribution, dimension, geometrical characteristics of the domain, position of the origin, etc.) on the spectral gap…
We provide error bounds on the traces and norms of the derivative of the $L^2$ projection of an $H^{k}$ function onto the space of polynomials of degree $\leq p$. The bounds are explicit in the order of differentiation and the polynomial…
We study the problem of approximating the total variation distance between two mixtures of product distributions over an $n$-dimensional discrete domain. Given two mixtures $\mathbb{P}$ and $\mathbb{Q}$ with $k_1$ and $k_2$ product…
Let X Nv(0, {\Lambda}) be a normal vector in v dimensions, where {\Lambda} is diagonal. With reference to the truncated distribution of X on the interior of a v-dimensional Euclidean ball, we completely prove a variance inequality and a…
Extending a result of Caffarelli, we provide global Lipschitz changes of variables between compactly supported perturbations of log-concave measures. The result is based on a combination of ideas from optimal transportation theory and a new…
In this short note, we develop a local approximation for the log-ratio of the multivariate hypergeometric probability mass function over the corresponding multinomial probability mass function. In conjunction with the bounds from Carter…
We investigate some previously unexplored (or underexplored) computational aspects of total variation (TV) distance. First, we give a simple deterministic polynomial-time algorithm for checking equivalence between mixtures of product…
We deal with stochastic differential equations with jumps. In order to obtain an accurate approximation scheme, it is usual to replace the "small jumps" by a Brownian motion. In this paper, we prove that for every fixed time $t$, the…