Related papers: The Geometry of Efficient Nonconvex Sampling

Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously.…

In the matter of selection of sample time points for the estimation of the power spectral density of a continuous time stationary stochastic process, irregular sampling schemes such as Poisson sampling are often preferred over regular…

In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and…

We consider the problem of recovering linear image of unknown signal belonging to a given convex compact signal set from noisy observation of another linear image of the signal. We develop a simple generic efficiently computable nonlinear…

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an…

We study differentially private (DP) optimization algorithms for stochastic and empirical objectives which are neither smooth nor convex, and propose methods that return a Goldstein-stationary point with sample complexity bounds that…

We present a new approach to learning the structure and parameters of a Bayesian network based on regularized estimation in an exponential family representation. Here we show that, given a fixed variable order, the optimal structure and…

This work explores a novel perspective on solving nonconvex and nonsmooth optimization problems by leveraging sampling based methods. Instead of treating the objective function purely through traditional (often deterministic) optimization…

In this paper, we consider the problem of identifying a linear map from measurements which are subject to intermittent and arbitarily large errors. This is a fundamental problem in many estimation-related applications such as fault…

In simulations of crystals, unlike liquids or gases, it may happen that the properties of the studied system depend not only on the volume of the simulation cell but also on its shape. For such cases it is desirable to change the shape of…

We present a method to obtain upper bounds on covering numbers. As applications of this method, we reprove and generalize results of Rogers on economically covering Euclidean $n$-space with translates of a convex body, or more generally,…

Let a random simplex in a d-dimensional convex body be the convex hull of d+1 random points from the body. We study the following question: As a function of the convex body, is the expected volume of a random simplex monotone non-decreasing…

In this work, we study how to use sampling to speed up mechanisms for answering adaptive queries into datasets without reducing the accuracy of those mechanisms. This is important to do when both the datasets and the number of queries asked…

We study sampling problems associated with potentials that lack smoothness. The potentials can be either convex or non-convex. Departing from the standard smooth setting, the potentials are only assumed to be weakly smooth or non-smooth, or…

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let X be an isotropic random vector in R^n with a log-concave density. For a typical subspace E in R^n of dimension n^c, consider the…

We present a mathematical and algorithmic scheme for learning the principal geometric elements in an image or 3D object. We build on recent work that convexifies the basic problem of finding a combination of a small number shapes that…

We study the problem of sampling from a $d$-dimensional distribution with density $p(x)\propto e^{-f(x)}$, which does not necessarily satisfy good isoperimetric conditions. Specifically, we show that for any $L,M$ satisfying $LM\ge d\ge 5$,…

We revisit the classical problem of finding an approximately stationary point of the average of $n$ smooth and possibly nonconvex functions. The optimal complexity of stochastic first-order methods in terms of the number of gradient…

This paper develops a unified framework for estimating the volume of a set in $\mathbb{R}^d$ based on observations of points uniformly distributed over the set. The framework applies to all classes of sets satisfying one simple axiom: a…

Sampling from circular distributions is a fundamental task in directional statistics. A key challenge in acceptance-rejection methods lies in selecting an efficient envelope density, as poor choices can lead to low acceptance rates and…