Efficient sampling from the Bingham distribution

• URL: http://arxiv.org/abs/2010.00137v2
• Date: Fri, 8 Dec 2023 19:43:42 GMT
• Title: Efficient sampling from the Bingham distribution
• Authors: Rong Ge, Holden Lee, Jianfeng Lu, Andrej Risteski
• Abstract summary: We give an exact sampling algorithm from the Bingham distribution $p(x)propto exp(xtop A x)$ on the sphere $mathcal Sd-1$ with expected runtime of $operatornamepoly(d, lambda_max(A)-lambda_min(A))$. As a direct application, we use this to sample from the posterior distribution of a rank-1 matrix inference problem in time.
• Score: 38.50073658077009
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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