Efficient sampling from the Bingham distribution

TL;DR

This paper presents an exact, polynomial-time sampling algorithm for the Bingham distribution on spheres, improving over MCMC methods by providing precise samples through rejection sampling with polynomial approximations.

Contribution

The authors introduce a novel rejection sampling algorithm for the Bingham distribution that achieves exact sampling with polynomial runtime, unlike traditional MCMC methods.

Findings

Algorithm achieves exact samples from the Bingham distribution.

Runtime is polynomial in dimension and eigenvalue gap.

Application demonstrated in rank-1 matrix inference.

Abstract

We give a algorithm for exact sampling from the Bingham distribution $p(x)\propto \exp(x^\top A x)$ on the sphere $\mathcal S^{d-1}$ with expected runtime of $\operatorname{poly}(d, \lambda_{\max}(A)-\lambda_{\min}(A))$. The algorithm is based on rejection sampling, where the proposal distribution is a polynomial approximation of the pdf, and can be sampled from by explicitly evaluating integrals of polynomials over the sphere. Our algorithm gives exact samples, assuming exact computation of an inverse function of a polynomial. This is in contrast with Markov Chain Monte Carlo algorithms, which are not known to enjoy rapid mixing on this problem, and only give approximate samples. As a direct application, we use this to sample from the posterior distribution of a rank-1 matrix inference problem in polynomial time.