A Cubic Algorithm for Computing Gaussian Volume

TL;DR

This paper introduces a new randomized algorithm with cubic complexity for efficiently estimating Gaussian measures and sampling within convex sets, improving upon previous methods by a factor of n.

Contribution

The paper presents a novel cubic-time algorithm for Gaussian volume estimation and sampling that avoids isotropic transformation and uses advanced isoperimetry and walk techniques.

Findings

Complexity of integration is $O^*(n^3)$.

Sampling complexity is $O^*(n^3)$ for the first sample and $O^*(n^2)$ thereafter.

Algorithm improves efficiency over previous methods by a factor of n.

Abstract

We present randomized algorithms for sampling the standard Gaussian distribution restricted to a convex set and for estimating the Gaussian measure of a convex set, in the general membership oracle model. The complexity of integration is $O^*(n^3)$ while the complexity of sampling is $O^*(n^3)$ for the first sample and $O^*(n^2)$ for every subsequent sample. These bounds improve on the corresponding state-of-the-art by a factor of $n$. Our improvement comes from several aspects: better isoperimetry, smoother annealing, avoiding transformation to isotropic position and the use of the "speedy walk" in the analysis.