{"paper":{"title":"On the tensorization of the variational distance","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Aryeh Kontorovich","submitted_at":"2024-09-16T15:13:05Z","abstract_excerpt":"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 $||P^\\otimes_{1:n}-Q^\\otimes_{1:n}||$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.10368","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.10368/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}