关于混合分布的桥接
On Bridging Mixture Distributions
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中文总结 AI 辅助
本文针对混合概率测度的桥接问题，提出基于马尔可夫核的通用采样机制，证明其熵最优性，推导高斯混合等场景下的2-瓦瑟斯坦连续性界并进行数值验证。
中文摘要 AI 辅助
本文研究两个混合概率测度之间的桥接问题。具体而言，给定两个分量分布之间的马尔可夫核，我们提供一种从一个混合分布生成样本到另一个混合分布的通用机制。针对给定的参考空间和扩展状态空间，我们证明了该方法的熵最优性。由于应用该思路需要已知基础混合分布和马尔可夫核，而这类信息通常难以获取，因此我们考虑高斯混合分布与薛定谔桥的情况。我们证明了精确桥与基于ε-协方差膨胀近似的桥之间存在通用的2-瓦瑟斯坦连续性界，该结论依赖于对扰动里卡蒂映射的新型连续性分析。我们将结果应用于高斯混合分布、单个高斯分布的桥接，以及高斯参数和蒙日映射的经验估计。对于高斯混合分布，当参数使用期望最大化（Expectation-Maximization）算法估计时，在假设条件下且概率至少为1-10N⁻¹时，真实桥与近似桥之间2-瓦瑟斯坦距离的上界为O([(d log N / N)^(1/2){1 + (d log N / N)^(1/2)(ε⁻² + 1)} + ε²])，其中d、N分别为高斯分布的维度和经验样本数量；对于后两种情况，该上界的期望为O(d{(1 + ε⁻²)/(1 + N) + ε²})。我们还对这些界进行了数值研究。
英文摘要
The construction and the stability of bridges between mixture probability measures are investigated. Given a collection of Markov transitions bridging the components of two mixtures, together with a coupling of the mixture labels, we construct a bridge between the mixtures on an extended state space, and we show that this bridge inherits the entropic optimality of the component bridges. We then turn to Gaussian mixtures bridged by Gaussian Schrödinger bridges, whose parameters are generally unknown and must be estimated. We prove a $2$-Wasserstein continuity theorem between the exact bridge and its plug-in approximation. The analysis rests on a representation of the Riccati fixed point map in inverse coordinates; this map is globally $1$-Lipschitz and requires no matrix inversion. These results are applied to three estimation schemes, with $d$ the dimension and $N$ the number of samples. For Gaussian mixtures estimated by the Expectation-Maximization algorithm, the squared $2$-Wasserstein error is of order $(d\log N/N)^{1/2}+d^2\log N/N$ with probability at least $1-10N^{-1}$. For single Gaussian marginals estimated by their sample moments, it is of order $d^2/N$ in expectation. For the regularized empirical Monge map based on an $ε$-inflation of the sample covariance, the mean squared error is of order $d^2\{(1+ε^{-1})/N+ε^2\}$; as soon as $N$ is a sufficiently large multiple of $d$, it reduces to $d^2/N+d\,ε^2$, up to an exponentially small term. These estimates are illustrated numerically.
发表机构
- Centre de Recherche Inria Bordeaux Sud-Ouest（法国国家信息与自动化研究所波尔多西南研究中心）
- Computing & Mathematical Sciences Division, Mohamed Bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学计算与数学科学系）
- School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）
机构由 AI 辅助整理，请以论文原文为准。
