The total variation distance between high-dimensional Gaussians with the same mean
Luc Devroye McGill University Supported by NSERC Grant A3456. Abbas Mehrabian McGill University Supported by an IVADO-Apogée-CFREF Postdoctoral Fellowship. Email: Tommy Reddad McGill University Supported by NSERC PGS D Scholarship 396164433.
Abstract
Given two high-dimensional Gaussians with the same mean, we prove a lower and an upper bound for their total variation distance, which are within a constant factor of one another.111In an earlier version, tight bounds were claimed for the total variation distance between two general Gaussians. But the proof of the upper bound was incorrect, and we removed the flawed bound from the paper. Later, Arbas, Ashtiani, and Liaw ([1, Theorem 1.8]) proved tight bounds for the total-variation distance between two general Gaussians, solving the original problem.
中文速览
两个高维高斯分布（Gaussian distribution）之间到底有多"不同"，是统计学和机器学习中的基本问题，但其全变差距离（total variation distance）没有解析闭合公式，长期缺乏紧的估计。这篇论文针对均值相同的两个高维高斯分布，证明了全变差距离的下界和上界，两者之间仅差一个常数倍，从而给出了实质上紧的刻画；关键量是矩阵 $\Sigma_1^{-1}\Sigma_2 - I$ 的特征值平方和的平方根，即两个协方差矩阵"差异程度"的 Frobenius 范数式度量。对于均值不同的情形，文章给出了一个下界并将上界留作公开问题（该问题随后被其他研究者解决）。这一结果为比较两个高斯分布提供了简洁、可计算的闭合形式估计，对高维统计推断、分布学习等应用具有直接价值。
原文 arXiv:1810.08693;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1810.08693v7