{"title":"On the tensorization of the variational distance","authors":"Aryeh Kontorovich","doi":"arxiv-2409.10368","DOIUrl":null,"url":null,"abstract":"If one seeks to estimate the total variation between two product measures\n$||P^\\otimes_{1:n}-Q^\\otimes_{1:n}||$ in terms of their marginal TV sequence\n$\\delta=(||P_1-Q_1||,||P_2-Q_2||,\\ldots,||P_n-Q_n||)$, then trivial upper and\nlower bounds are provided by$ ||\\delta||_\\infty \\le\n||P^\\otimes_{1:n}-Q^\\otimes_{1:n}||\\le||\\delta||_1$. We improve the lower bound\nto $||\\delta||_2\\lesssim||P^\\otimes_{1:n}-Q^\\otimes_{1:n}||$, thereby reducing\nthe gap between the upper and lower bounds from $\\sim n$ to $\\sim\\sqrt $.\nFurthermore, we show that {\\em any} estimate on\n$||P^\\otimes_{1:n}-Q^\\otimes_{1:n}||$ expressed in terms of $\\delta$ must\nnecessarily exhibit a gap of $\\sim\\sqrt n$ between the upper and lower bounds\nin the worst case, establishing a sense in which our estimate is optimal.\nFinally, we identify a natural class of distributions for which $||\\delta||_2$\napproximates the TV distance up to absolute multiplicative constants.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10368","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.