Related papers: Gaussian Approximation of Convex Sets by Intersect…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
Gaussian mixture distributions are commonly employed to represent general probability distributions. Despite the importance of using Gaussian mixtures for uncertainty estimation, the entropy of a Gaussian mixture cannot be calculated…
Many star bodies have convex subsets with approximately the same Gaussian measure (of the complement). Inspired by this phenomenon, and in connection with the randomized Dvoretzky theorem for Lorentz spaces, we derive bounds on the…
This paper investigates the approximation of Gaussian random variables in Banach spaces, focusing on the high-probability bounds for the approximation of Gaussian random variables using finitely many observations. We derive non-asymptotic…
In this paper we provide explicit upper bounds on some distances between the (law of the) output of a random Gaussian NN and (the law of) a random Gaussian vector. Our results concern both shallow random Gaussian neural networks with…
The Gaussian Correlation Conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the…
Approximating complex probability distributions, such as Bayesian posterior distributions, is of central interest in many applications. We study the expressivity of geometric Gaussian approximations. These consist of approximations by…
Polyhedral-type approximations of convex-like domains in $\mathbb{C}^d$ have been considered recently by the second author. In particular, the decay rate of the error in optimal volume approximation as a function of the number of facets has…
We study sequential probability assignment in the Gaussian setting, where the goal is to predict, or equivalently compress, a sequence of real-valued observations almost as well as the best Gaussian distribution with mean constrained to a…
This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the…
This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space,…
We study a general smallest intersecting ball problem and its soft-margin variant in high-dimensional Euclidean spaces for input objects that are compact and convex. These two problems link and unify a series of fundamental problems in…
Conventional approximations to Bayesian inference rely on either approximations by statistics such as mean and covariance or by point particles. Recent advances such as the ensemble Gaussian mixture filter have generalized these notions to…
A popular class of problem in statistics deals with estimating the support of a density from $n$ observations drawn at random from a $d$-dimensional distribution. The one-dimensional case reduces to estimating the end points of a univariate…
This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for…
We present an efficient algorithm to compute tight upper bounds of collision probability between two objects with positional uncertainties, whose error distributions are represented with non-Gaussian forms. Our approach can handle noisy…
This paper concerns a method of testing equality of distribution of random convex compact sets and the way how to use the test to distinguish between two realisations of general random sets. The family of metrics on the space of…
Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance…
The Haugazeau method was originally designed to compute the best approximation from an intersection of closed convex sets in Hilbert spaces using the projection operators onto the individual sets iteratively. We propose an abstract…
Generalization error bounds are essential to understanding machine learning algorithms. This paper presents novel expected generalization error upper bounds based on the average joint distribution between the output hypothesis and each…