发表机构
University of Oxford; UPC; University of Toronto（牛津大学; 加泰罗尼亚理工大学; 多伦多大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对带薄膜先验的高斯图模型，通过对偶正态因子图加快随机扫描吉布斯采样的收敛速度，建立了原始与对偶模型协方差关系，实验验证了理论结果。
AI 中文摘要
我们研究了带有薄膜先验的高斯图模型的随机扫描吉布斯采样器的收敛性质。我们证明，在对偶模型中，吉布斯采样器的收敛速度显著加快，对偶模型是通过对表示原始模型的正态因子图的局部因子应用傅里叶变换得到的。在两个域中，我们推导了齐次k-正则图的精确收敛速度。我们证明，对于所有其图表示包含环的齐次模型，对偶域中的收敛速度是通用的，且与底层图拓扑无关。此外，我们表明对偶域中的有效收敛速度由图的代数连通性决定，在不增加每次扫描计算复杂度的情况下提供额外加速。我们进一步建立了原始模型和对偶模型协方差结构之间的显式代数关系，使得可以直接从对偶模型的边际统计量恢复原始模型的边际统计量。最后，对多个图族的数值实验证实了我们的理论结果，并证明在各种设置下收敛速度有显著提升。
英文摘要
We study the convergence properties of the random-sweep Gibbs sampler for Gaussian graphical models with a thin-membrane prior. We demonstrate that the convergence rate of the Gibbs sampler is significantly accelerated in the dual model, which is obtained by applying the Fourier transform to the local factors of the normal factor graph representing the original model. In both domains, we derive the exact convergence rates for homogeneous $k$-regular graphs. We prove that, for all homogeneous models whose graphical representations contain cycles, the convergence rate in the dual domain is universal and independent of the underlying graph topology. Moreover, we show that the effective convergence rate in the dual domain is governed by the algebraic connectivity of the graph, providing an additional acceleration without increasing the computational complexity per sweep. We further establish an explicit algebraic relation between the covariance structures of the primal and dual models, enabling marginal statistics of the primal model to be recovered directly from those of the dual model. Finally, numerical experiments on several graph families confirm our theoretical results and demonstrate substantial improvements in the convergence rates in various settings.
