Efficient sampling from the Bingham distribution
This provides an efficient exact sampling method for a specific distribution in statistics and machine learning, addressing a bottleneck where existing MCMC methods are slow or approximate.
The authors tackled the problem of exact sampling from the Bingham distribution on a sphere, achieving an algorithm with expected runtime polynomial in dimension and eigenvalue gap, and applied it to sample from a rank-1 matrix inference posterior in polynomial time.
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 $\operatorname{poly}(d, λ_{\max}(A)-λ_{\min}(A))$. 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.