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.

翻译：我们提出了一种从球面$\mathcal S^{d-1}$上的Bingham分布$p(x)\propto \exp(x^\top A x)$中进行精确采样的算法，其期望运行时间为$\operatorname{poly}(d, \lambda_{\max}(A)-\lambda_{\min}(A))$。该算法基于拒绝采样，其中提议分布是概率密度函数的多项式近似，可通过显式计算球面上多项式的积分进行采样。该算法在假设能精确计算多项式反函数的前提下，可生成精确样本。这与马尔可夫链蒙特卡洛算法形成对比——后者在该问题上尚未被证明具有快速混合性，且仅能提供近似采样。作为直接应用，我们利用该算法在多项式时间内实现了秩1矩阵推断问题后验分布的采样。