宾厄姆分布的Kent-Ganeiber-Mardia采样器的复杂度边界
A Complexity Bound for the Kent-Ganeiber-Mardia Sampler for the Bingham Distribution
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中文总结 AI 辅助
本文针对宾厄姆分布的Kent-Ganeiber-Mardia拒绝采样器，验证了其高浓度极限下接受概率的d^(-1/2)阶复杂度边界，证明了该速率不可改进并给出了常数下界。
中文摘要 AI 辅助
宾厄姆分布是单位球面上一类关于对径点对称的分布，其特征是相对于均匀分布的测度为二次型的指数形式。Kent、Ganeiber和Mardia提出了一种拒绝采样器，用于从宾厄姆分布中生成样本，该采样器使用角中心高斯（ACG）分布作为建议分布。他们的实验结果表明，效率最低的情况是高浓度极限，此时接受概率在维度d下的阶为d^(-1/2)，这意味着存在多项式复杂度保证。在本研究中，我们验证了这种依赖维度的预测，建立了统一的保证：对于所有D=D^T∈ℝ^(d×d)，inf{α_D}≥c_*/√d，其中c_*=0.759…。一维高浓度极限表明，d^(-1/2)的速率是不可改进的，甚至常数c_*也无法超过0.857…。该证明依赖于对接受概率的新颖解释以及卡方随机变量加权和的比较原理，这可能具有独立的研究价值。
英文摘要
The Bingham distribution is a family of antipodally symmetric distributions on the unit sphere, characterised by an exponential-of-quadratic change of measure with respect to the uniform distribution. Kent, Ganeiber and Mardia proposed a rejection sampler for generating samples from Bingham distributions using proposals from an angular central Gaussian (ACG) distribution. Their empirical results suggest that the least efficient regime is the high-concentration limit, where the acceptance probability is of order $d^{-1/2}$ in dimension $d$, implying a polynomial complexity guarantee. In this note, we verify this dimension-dependent prediction, establishing the uniform guarantee $\inf\{α_D:D=D^\top\in\mathbb{R}^{d\times d}\}\ge c_\star/\sqrt{d}$, where $c_\star=0.759\ldots$. A one-dimensional high-concentration limit demonstrates that the $d^{-1/2}$ rate is unimprovable and that even the constant $c_\star$ cannot be improved beyond $0.857\ldots$. The proof relies on a novel interpretation of the acceptance probability and a comparison principle for weighted sums of chi-squared random variables, which may be of independent interest.
