The total variation distance between high-dimensional Gaussians with the same mean

摘要

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.11 1 In 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.