{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:LEWONTVEZN3RG45FSJ7OS7MXIU","short_pith_number":"pith:LEWONTVE","schema_version":"1.0","canonical_sha256":"592ce6cea4cb771373a5927ee97d97452e98259ef20698c3a5163652407f219e","source":{"kind":"arxiv","id":"2107.06224","version":2},"attestation_state":"computed","paper":{"title":"T product Tensors Part II: Tail Bounds for Sums of Random T product Tensors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Shih Yu Chang, Yimin Wei","submitted_at":"2021-07-13T16:23:51Z","abstract_excerpt":"This paper is the Part II of a serious work about T product tensors focusing at establishing new probability bounds for sums of random, independent, T product tensors. These probability bounds characterize large deviation behavior of the extreme eigenvalue of the sums of random T product tensors. We apply Lapalace transform method and Lieb concavity theorem for T product tensors obtained from our Part I paper, and apply these tools to generalize the classical bounds associated with the names Chernoff, and Bernstein from the scalar to the T product tensor setting. Tail bounds for the norm of a "},"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":"2107.06224","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2021-07-13T16:23:51Z","cross_cats_sorted":[],"title_canon_sha256":"801ec2b8d24d22842e9e2385056582e3baee6771b932882866c298832ca33b8d","abstract_canon_sha256":"b99f674b00955788d01b13b57ff0b95b10fb1817ed8a0c9a01c5ac6a18b83bc3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:39:15.890474Z","signature_b64":"y9BJVY/vfKAImZSTaHlEk+SLdLQ6bj66AqpT9SgDjyQ38TRlq3xtgustQe3I7WrdJZ61BzBo+EzEyXIMkxIaAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"592ce6cea4cb771373a5927ee97d97452e98259ef20698c3a5163652407f219e","last_reissued_at":"2026-07-05T03:39:15.890012Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:39:15.890012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"T product Tensors Part II: Tail Bounds for Sums of Random T product Tensors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Shih Yu Chang, Yimin Wei","submitted_at":"2021-07-13T16:23:51Z","abstract_excerpt":"This paper is the Part II of a serious work about T product tensors focusing at establishing new probability bounds for sums of random, independent, T product tensors. These probability bounds characterize large deviation behavior of the extreme eigenvalue of the sums of random T product tensors. We apply Lapalace transform method and Lieb concavity theorem for T product tensors obtained from our Part I paper, and apply these tools to generalize the classical bounds associated with the names Chernoff, and Bernstein from the scalar to the T product tensor setting. Tail bounds for the norm of a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.06224","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/2107.06224/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":"2107.06224","created_at":"2026-07-05T03:39:15.890064+00:00"},{"alias_kind":"arxiv_version","alias_value":"2107.06224v2","created_at":"2026-07-05T03:39:15.890064+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.06224","created_at":"2026-07-05T03:39:15.890064+00:00"},{"alias_kind":"pith_short_12","alias_value":"LEWONTVEZN3R","created_at":"2026-07-05T03:39:15.890064+00:00"},{"alias_kind":"pith_short_16","alias_value":"LEWONTVEZN3RG45F","created_at":"2026-07-05T03:39:15.890064+00:00"},{"alias_kind":"pith_short_8","alias_value":"LEWONTVE","created_at":"2026-07-05T03:39:15.890064+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.21667","citing_title":"Operator Inequalities in $\\Phi$-Product Tensor Algebras: Invariance and Transform Sensitivity","ref_index":10,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LEWONTVEZN3RG45FSJ7OS7MXIU","json":"https://pith.science/pith/LEWONTVEZN3RG45FSJ7OS7MXIU.json","graph_json":"https://pith.science/api/pith-number/LEWONTVEZN3RG45FSJ7OS7MXIU/graph.json","events_json":"https://pith.science/api/pith-number/LEWONTVEZN3RG45FSJ7OS7MXIU/events.json","paper":"https://pith.science/paper/LEWONTVE"},"agent_actions":{"view_html":"https://pith.science/pith/LEWONTVEZN3RG45FSJ7OS7MXIU","download_json":"https://pith.science/pith/LEWONTVEZN3RG45FSJ7OS7MXIU.json","view_paper":"https://pith.science/paper/LEWONTVE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2107.06224&json=true","fetch_graph":"https://pith.science/api/pith-number/LEWONTVEZN3RG45FSJ7OS7MXIU/graph.json","fetch_events":"https://pith.science/api/pith-number/LEWONTVEZN3RG45FSJ7OS7MXIU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LEWONTVEZN3RG45FSJ7OS7MXIU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LEWONTVEZN3RG45FSJ7OS7MXIU/action/storage_attestation","attest_author":"https://pith.science/pith/LEWONTVEZN3RG45FSJ7OS7MXIU/action/author_attestation","sign_citation":"https://pith.science/pith/LEWONTVEZN3RG45FSJ7OS7MXIU/action/citation_signature","submit_replication":"https://pith.science/pith/LEWONTVEZN3RG45FSJ7OS7MXIU/action/replication_record"}},"created_at":"2026-07-05T03:39:15.890064+00:00","updated_at":"2026-07-05T03:39:15.890064+00:00"}