{"title":"关于变分距离的张量化","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":"{\"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}","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}
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.