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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 $||P^\\otimes_{1:n}-Q^\\otimes_{1:n}||$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2409.10368","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2024-09-16T15:13:05Z","cross_cats_sorted":[],"title_canon_sha256":"ad5b703f3982f5e65a67cc41bb5049abc9c077ae480fe1f0aa95946d0e86e0ad","abstract_canon_sha256":"c33a2ca671e37dd49298a89c3b5db7f78f371492e9dab9a90c65c251c823ea90"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:14:38.431524Z","signature_b64":"miTOHHJhCJMI4puiALmYuWplxsyn4Eg7N2B+RMvOALmcGWzfDXgBdi1JKlUenaUe+cQY2HC80hPgBY1Q9Q4hAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29035959e789781ddb70ffd1f7b53cfd7961a90496fcfdd299caa75a1b13f3c0","last_reissued_at":"2026-07-05T09:14:38.430967Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:14:38.430967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2409.10368","created_at":"2026-07-05T09:14:38.431053+00:00"},{"alias_kind":"arxiv_version","alias_value":"2409.10368v2","created_at":"2026-07-05T09:14:38.431053+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.10368","created_at":"2026-07-05T09:14:38.431053+00:00"},{"alias_kind":"pith_short_12","alias_value":"FEBVSWPHRF4B","created_at":"2026-07-05T09:14:38.431053+00:00"},{"alias_kind":"pith_short_16","alias_value":"FEBVSWPHRF4B3W3Q","created_at":"2026-07-05T09:14:38.431053+00:00"},{"alias_kind":"pith_short_8","alias_value":"FEBVSWPH","created_at":"2026-07-05T09:14:38.431053+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.03839","citing_title":"On Computing Total Variation Distance Between Mixtures of Product Distributions","ref_index":34,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FEBVSWPHRF4B3W3Q77I7PNJ47V","json":"https://pith.science/pith/FEBVSWPHRF4B3W3Q77I7PNJ47V.json","graph_json":"https://pith.science/api/pith-number/FEBVSWPHRF4B3W3Q77I7PNJ47V/graph.json","events_json":"https://pith.science/api/pith-number/FEBVSWPHRF4B3W3Q77I7PNJ47V/events.json","paper":"https://pith.science/paper/FEBVSWPH"},"agent_actions":{"view_html":"https://pith.science/pith/FEBVSWPHRF4B3W3Q77I7PNJ47V","download_json":"https://pith.science/pith/FEBVSWPHRF4B3W3Q77I7PNJ47V.json","view_paper":"https://pith.science/paper/FEBVSWPH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2409.10368&json=true","fetch_graph":"https://pith.science/api/pith-number/FEBVSWPHRF4B3W3Q77I7PNJ47V/graph.json","fetch_events":"https://pith.science/api/pith-number/FEBVSWPHRF4B3W3Q77I7PNJ47V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FEBVSWPHRF4B3W3Q77I7PNJ47V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FEBVSWPHRF4B3W3Q77I7PNJ47V/action/storage_attestation","attest_author":"https://pith.science/pith/FEBVSWPHRF4B3W3Q77I7PNJ47V/action/author_attestation","sign_citation":"https://pith.science/pith/FEBVSWPHRF4B3W3Q77I7PNJ47V/action/citation_signature","submit_replication":"https://pith.science/pith/FEBVSWPHRF4B3W3Q77I7PNJ47V/action/replication_record"}},"created_at":"2026-07-05T09:14:38.431053+00:00","updated_at":"2026-07-05T09:14:38.431053+00:00"}