Topics in Statistics and Convex Geometry: Rounding, Sampling, and Interpolation

Abstract

We consider a few aspects of the interplay between convex geometry and statistics. We consider three problems of interest: how to bring a convex body specified by a self-concordant barrier into a suitably “rounded” position using an affine-invariant random walk; how to design a rapidly-mixing affine-invariant random walk with maximal volume ellipsoids; and how to perform interpolation in multiple dimensions given noisy observations of the original function on a finite set. We begin with an overview of some background information on convex bodies which recur in this dissertation, discussing polytopes, the Dikin ellipsoid, and John’s ellipsoid in particular. We also discuss cutting plane methods and Vaidya’s algorithm, which we employ in subsequent analysis. We then review Markov chains on general state spaces to motivate designing rapidly mixing geometric random walks, and the means by which the mixing time may be analyzed. Using these results regarding convex bodies and general state space Markov chains, along with recently developed concentration inequalities on general state space Markov chains, we employ an affine-invariant random walk using Dikin ellipsoids to provably bring a convex body specified by a self-concordant barrier into an approximately “rounded” position. We also design a random walk using John’s ellipsoids and derive its mixing time. Departing somewhat from these themes, we also discuss regression in multiple dimensions over classes of continuously differentiable functions. We provide a cutting plane algorithm for the case of first-order differentiable functions with Lipschitz gradients. We additionally consider the more general case of higher-order continuously differentiable functions, proving Whitney’s extension theorem for this case, and outlining a quadratic program to perform regression for this setting.