{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:AF7XMA22ODPO2S3OFGQ34OA2JA","short_pith_number":"pith:AF7XMA22","canonical_record":{"source":{"id":"1004.4389","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-04-25T23:21:23Z","cross_cats_sorted":[],"title_canon_sha256":"92e12765405b869d8b99cadc928246ded5806d496fdf50d7eb230913e3aba534","abstract_canon_sha256":"567d63288ab21f5b795f9480f747cbef4e6ef4843473b7348c2918e064b390f1"},"schema_version":"1.0"},"canonical_sha256":"017f76035a70deed4b6e29a1be381a480f5f40aa7f5c7d1d4a575802d43a91ad","source":{"kind":"arxiv","id":"1004.4389","version":7},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.4389","created_at":"2026-05-18T02:53:15Z"},{"alias_kind":"arxiv_version","alias_value":"1004.4389v7","created_at":"2026-05-18T02:53:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.4389","created_at":"2026-05-18T02:53:15Z"},{"alias_kind":"pith_short_12","alias_value":"AF7XMA22ODPO","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"AF7XMA22ODPO2S3O","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"AF7XMA22","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:AF7XMA22ODPO2S3OFGQ34OA2JA","target":"record","payload":{"canonical_record":{"source":{"id":"1004.4389","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-04-25T23:21:23Z","cross_cats_sorted":[],"title_canon_sha256":"92e12765405b869d8b99cadc928246ded5806d496fdf50d7eb230913e3aba534","abstract_canon_sha256":"567d63288ab21f5b795f9480f747cbef4e6ef4843473b7348c2918e064b390f1"},"schema_version":"1.0"},"canonical_sha256":"017f76035a70deed4b6e29a1be381a480f5f40aa7f5c7d1d4a575802d43a91ad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:15.984571Z","signature_b64":"fwBeLHg7D5QAhVEGYAP2pHfvQg4vGsuV6BbiQW7D0p2p9V3JpgM7OZuotfl/T31wlWmknjhs+HZSszBIBlPQBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"017f76035a70deed4b6e29a1be381a480f5f40aa7f5c7d1d4a575802d43a91ad","last_reissued_at":"2026-05-18T02:53:15.984019Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:15.984019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.4389","source_version":7,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:53:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tcvhtbR8lyiut/zcV00RWN9LVw1+a5SSDdkRcC04ZH5F8yLegIPPiped/ga1BC7HnD+nLj/veL40cbdHiqW8BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T12:55:34.371869Z"},"content_sha256":"821344f462b1f0d8e652e54945d5f925ad054cd40218fca4e0d2d2cb268463ba","schema_version":"1.0","event_id":"sha256:821344f462b1f0d8e652e54945d5f925ad054cd40218fca4e0d2d2cb268463ba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:AF7XMA22ODPO2S3OFGQ34OA2JA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"User-friendly tail bounds for sums of random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Joel A. Tropp","submitted_at":"2010-04-25T23:21:23Z","abstract_excerpt":"This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales.\n  In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Be"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.4389","kind":"arxiv","version":7},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:53:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FPGHsBupGO9c8Ze8d9OOJTsgWd39fwOfuN5ouP0phCcsn9VlqAwhj2pOR0/ZlujtMaHST/NCMrCifUcJrpQAAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T12:55:34.372568Z"},"content_sha256":"581ad4c88c728caddd516b67ea6b1499a79adbc6f1e0f27d3397dd0b7953290e","schema_version":"1.0","event_id":"sha256:581ad4c88c728caddd516b67ea6b1499a79adbc6f1e0f27d3397dd0b7953290e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AF7XMA22ODPO2S3OFGQ34OA2JA/bundle.json","state_url":"https://pith.science/pith/AF7XMA22ODPO2S3OFGQ34OA2JA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AF7XMA22ODPO2S3OFGQ34OA2JA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T12:55:34Z","links":{"resolver":"https://pith.science/pith/AF7XMA22ODPO2S3OFGQ34OA2JA","bundle":"https://pith.science/pith/AF7XMA22ODPO2S3OFGQ34OA2JA/bundle.json","state":"https://pith.science/pith/AF7XMA22ODPO2S3OFGQ34OA2JA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AF7XMA22ODPO2S3OFGQ34OA2JA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:AF7XMA22ODPO2S3OFGQ34OA2JA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"567d63288ab21f5b795f9480f747cbef4e6ef4843473b7348c2918e064b390f1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-04-25T23:21:23Z","title_canon_sha256":"92e12765405b869d8b99cadc928246ded5806d496fdf50d7eb230913e3aba534"},"schema_version":"1.0","source":{"id":"1004.4389","kind":"arxiv","version":7}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.4389","created_at":"2026-05-18T02:53:15Z"},{"alias_kind":"arxiv_version","alias_value":"1004.4389v7","created_at":"2026-05-18T02:53:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.4389","created_at":"2026-05-18T02:53:15Z"},{"alias_kind":"pith_short_12","alias_value":"AF7XMA22ODPO","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"AF7XMA22ODPO2S3O","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"AF7XMA22","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:581ad4c88c728caddd516b67ea6b1499a79adbc6f1e0f27d3397dd0b7953290e","target":"graph","created_at":"2026-05-18T02:53:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales.\n  In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Be","authors_text":"Joel A. Tropp","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-04-25T23:21:23Z","title":"User-friendly tail bounds for sums of random matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.4389","kind":"arxiv","version":7},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:821344f462b1f0d8e652e54945d5f925ad054cd40218fca4e0d2d2cb268463ba","target":"record","created_at":"2026-05-18T02:53:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"567d63288ab21f5b795f9480f747cbef4e6ef4843473b7348c2918e064b390f1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-04-25T23:21:23Z","title_canon_sha256":"92e12765405b869d8b99cadc928246ded5806d496fdf50d7eb230913e3aba534"},"schema_version":"1.0","source":{"id":"1004.4389","kind":"arxiv","version":7}},"canonical_sha256":"017f76035a70deed4b6e29a1be381a480f5f40aa7f5c7d1d4a575802d43a91ad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"017f76035a70deed4b6e29a1be381a480f5f40aa7f5c7d1d4a575802d43a91ad","first_computed_at":"2026-05-18T02:53:15.984019Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:53:15.984019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fwBeLHg7D5QAhVEGYAP2pHfvQg4vGsuV6BbiQW7D0p2p9V3JpgM7OZuotfl/T31wlWmknjhs+HZSszBIBlPQBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:53:15.984571Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.4389","source_kind":"arxiv","source_version":7}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:821344f462b1f0d8e652e54945d5f925ad054cd40218fca4e0d2d2cb268463ba","sha256:581ad4c88c728caddd516b67ea6b1499a79adbc6f1e0f27d3397dd0b7953290e"],"state_sha256":"8404376ecb542ba2ade5f40340efddc60a0df98f5f7bb614db39f13eca85c8a4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TEG14mhmKQkiiIpI1KYCmQ8OAyX8ODYeh/VU1IieMe245aWTG/mZAy/iI01u9lqk5MbbMohRzsfAVxf82MUMDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T12:55:34.376047Z","bundle_sha256":"c6a5e35934949645421b6574fac9fabaebd8eb3580455d50e3c754480fa9ad81"}}