Related papers: Push forward measures and concentration phenomena

The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the…

We show that in any complete metric space the probability measures $\mu$ with compact and connected support are the ones having the property that the optimal tranportation distance to any other probability measure $\nu$ living on the…

Many recent works have shown that adversarial examples that fool classifiers can be found by minimally perturbing a normal input. Recent theoretical results, starting with Gilmer et al. (2018b), show that if the inputs are drawn from a…

Let $\mu$ be a log-concave probability measure on ${\mathbb R}^n$ and for any $N>n$ consider the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$, where $X_1,X_2,\ldots $ are independent random points in ${\mathbb R}^n$ distributed…

The influence theorem for product measures on the discrete space {0,1}^N may be extended to probability measures with the property of monotonicity (which is equivalent to `strong positive-association'). Corresponding results are valid for…

By using the spectrum of the underlying symmetric diffusion operator, the convergence in $L^p$-Wasserstein distance $\mathbb W_p (p\ge 1)$ is characterized for the empirical measure $\mu_t$ of non-symmetric subordinated diffusion processes…

We give a concentration inequality based on the premise that random variables take values within a particular region. The concentration inequality guarantees that, for any sequence of correlated random variables, the difference between the…

In this paper, we consider the well known problem of estimating a density function under qualitative assumptions. More precisely, we estimate monotone non increasing densities in a Bayesian setting and derive concentration rate for the…

We present vector-valued concentration inequalities for the biased measure on the discrete hypercube with an optimal dependence on the bias parameter and the Rademacher type of the target Banach space. These results allow us to obtain novel…

Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where Lipschitz is taken with respect to the metric induced by a weighted covering of the index set…

Random measures provide flexible parameters for Bayesian nonparametric models. Given two different priors for a random measure, we develop a natural framework to investigate the rate at which the corresponding posteriors merge, as the…

We establish a general concentration result for the 1-Wasserstein distance between the empirical measure of a sequence of random variables and its expectation. Unlike standard results that rely on independence (e.g., Sanov's theorem) or…

Given a Radon probability measure $\mu$ supported in $\mathbb{R}^d$, we are interested in those points $x$ around which the measure is concentrated infinitely many times on thin annuli centered at $x$. Depending on the lower and upper…

In this paper we propose a method to construct probability measures on the space of convex bodies with a given pushforward distribution. Concretely we show that there is a measure on the metric space of centrally symmetric convex bodies,…

We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.

Similar to pure fluids, physical properties of a binary fluid mixture also exhibit singularities close to its critical point, especially when it is dilute. The numerical and theoretical results presented in this paper identify that the…

The distribution $\mu_{cl}$ of a Poisson cluster process in $X=\mathbb{R}^{d}$ (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in $\mathfrak{X}=\sqcup_{n} X^n$, with intensity measure…

This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $\mu_i$ and $\nu_i, i=1,2$, are close to each other in terms of the L\'{e}vy metric and if the…

Proximity measurements probe whether pairs of particles are close to one another. We consider the impact of post-selected random proximity measurements on a quantum fluid of many distinguishable particles. We show that such measurements…

We consider a Markov chain on $\mathbb{R}^d$ with invariant measure $\mu$. We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to various dual distances, including in particular…