On the tensorization of the variational distance

TL;DR

This paper improves bounds on estimating the total variation distance between product measures using marginal TV sequences, reducing the gap from linear to square-root scale, and identifies cases where the approximation is tight.

Contribution

It introduces a tighter lower bound using the Euclidean norm and proves the optimality of this estimate in the worst case.

Findings

Lower bound improved to ||δ||_2, reducing the gap to √n.

Any estimate based on δ must have a gap of order √n in the worst case.

Identifies distribution classes where ||δ||_2 approximates the total variation distance.

Abstract

If one seeks to estimate the total variation between two product measures $||P^\otimes_{1:n}-Q^\otimes_{1:n}||$ in terms of their marginal TV sequence $\delta=(||P_1-Q_1||,||P_2-Q_2||,\ldots,||P_n-Q_n||)$, then trivial upper and lower bounds are provided by$ ||\delta||_\infty \le ||P^\otimes_{1:n}-Q^\otimes_{1:n}||\le||\delta||_1$. We improve the lower bound to $||\delta||_2\lesssim||P^\otimes_{1:n}-Q^\otimes_{1:n}||$, thereby reducing the gap between the upper and lower bounds from $\sim n$ to $\sim\sqrt $. Furthermore, we show that {\em any} estimate on $||P^\otimes_{1:n}-Q^\otimes_{1:n}||$ expressed in terms of $\delta$ must necessarily exhibit a gap of $\sim\sqrt n$ between the upper and lower bounds in the worst case, establishing a sense in which our estimate is optimal. Finally, we identify a natural class of distributions for which $||\delta||_2$ approximates the TV distance up to absolute multiplicative constants.