Reverse Euclidean and Gaussian isoperimetric inequalities for parallel sets with applications

TL;DR

This paper establishes upper bounds on the surface area of parallel sets in Euclidean and Gaussian spaces, introduces reverse inequalities for these sets, and applies these results to problems in machine learning such as complexity bounds and robust risk estimation.

Contribution

It provides new reverse isoperimetric inequalities for parallel sets and demonstrates their applications in machine learning theory.

Findings

Surface area bounds for $r$-parallel sets in Euclidean space

Reverse Brunn-Minkowski inequality for parallel sets

Applications to complexity and risk estimation in machine learning

Abstract

The $r$-parallel set of a measurable set $A \subseteq \mathbb R^d$ is the set of all points whose distance from $A$ is at most $r$. In this paper, we show that the surface area of an $r$-parallel set in $\mathbb R^d$ with volume at most $V$ is upper-bounded by $e^{\Theta(d)}V/r$, whereas its Gaussian surface area is upper-bounded by $\max(e^{\Theta(d)}, e^{\Theta(d)}/r)$. We also derive a reverse form of the Brunn-Minkowski inequality for $r$-parallel sets, and as an aside a reverse entropy power inequality for Gaussian-smoothed random variables. We apply our results to two problems in theoretical machine learning: (1) bounding the computational complexity of learning $r$-parallel sets under a Gaussian distribution; and (2) bounding the sample complexity of estimating robust risk, which is a notion of risk in the adversarial machine learning literature that is analogous to the Bayes risk in hypothesis testing.