{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JRYX654WKYFN6QPCC2RIXX4FTX","short_pith_number":"pith:JRYX654W","canonical_record":{"source":{"id":"1501.01571","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-07T17:46:02Z","cross_cats_sorted":["cs.DS","cs.IT","cs.NA","math.IT","stat.ML"],"title_canon_sha256":"88214327cf8b0cafeb017580ad05962c1da5fdecca8b19b9140d730578bdb27e","abstract_canon_sha256":"68c6edab3028b78aa9cfb659f138a3ecd1e6096b399fea957c6cba212556bb4d"},"schema_version":"1.0"},"canonical_sha256":"4c717f7796560adf41e216a28bdf859dc0983bb30c0f60058cc5a4a745df48a2","source":{"kind":"arxiv","id":"1501.01571","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.01571","created_at":"2026-05-18T02:29:53Z"},{"alias_kind":"arxiv_version","alias_value":"1501.01571v1","created_at":"2026-05-18T02:29:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.01571","created_at":"2026-05-18T02:29:53Z"},{"alias_kind":"pith_short_12","alias_value":"JRYX654WKYFN","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JRYX654WKYFN6QPC","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JRYX654W","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JRYX654WKYFN6QPCC2RIXX4FTX","target":"record","payload":{"canonical_record":{"source":{"id":"1501.01571","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-07T17:46:02Z","cross_cats_sorted":["cs.DS","cs.IT","cs.NA","math.IT","stat.ML"],"title_canon_sha256":"88214327cf8b0cafeb017580ad05962c1da5fdecca8b19b9140d730578bdb27e","abstract_canon_sha256":"68c6edab3028b78aa9cfb659f138a3ecd1e6096b399fea957c6cba212556bb4d"},"schema_version":"1.0"},"canonical_sha256":"4c717f7796560adf41e216a28bdf859dc0983bb30c0f60058cc5a4a745df48a2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:53.745311Z","signature_b64":"HqM09t3/m6sCGvm9plEOgYQXyAvtNhqZfLiMdpw1uA2YyxuLYrG2Lkkv4S3InJr1a0PZp8aCVzwtPHjW+Ir4CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c717f7796560adf41e216a28bdf859dc0983bb30c0f60058cc5a4a745df48a2","last_reissued_at":"2026-05-18T02:29:53.744853Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:53.744853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.01571","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:29:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4jkzV0/MfX5AyVLGOAEJsnw0CKjGx0wTN9yNieBEtWcjVUq49yqZxdIlGMZAX8+b7oSyuiN9qwAWA7P536b2AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T04:55:48.938480Z"},"content_sha256":"93cc0518abda4e7bbdf277f7d10ab907b08ad043b4e21ceb1f15c22020ff792e","schema_version":"1.0","event_id":"sha256:93cc0518abda4e7bbdf277f7d10ab907b08ad043b4e21ceb1f15c22020ff792e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JRYX654WKYFN6QPCC2RIXX4FTX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Introduction to Matrix Concentration Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","cs.IT","cs.NA","math.IT","stat.ML"],"primary_cat":"math.PR","authors_text":"Joel A. Tropp","submitted_at":"2015-01-07T17:46:02Z","abstract_excerpt":"In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point where we can conquer many (formerly) challenging problems with a page or two of arithmetic. The aim of this monograph is to describe the most successful methods from this area along with some interesting examples that these techniques can illuminate."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01571","kind":"arxiv","version":1},"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":"2026-05-18T02:29:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IdrA8c7e6KccA/wRdEtg0R60PXG54TkyF6qC/DfvR9VONj2GsjKDMydzMtGDOvU89d4al1xwpBJtSDPpmYh7CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T04:55:48.939201Z"},"content_sha256":"d26dcdbc57411390e598fb01c38e7c74c3adb5b3d56f9046fca4bf9a7cc30d43","schema_version":"1.0","event_id":"sha256:d26dcdbc57411390e598fb01c38e7c74c3adb5b3d56f9046fca4bf9a7cc30d43"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JRYX654WKYFN6QPCC2RIXX4FTX/bundle.json","state_url":"https://pith.science/pith/JRYX654WKYFN6QPCC2RIXX4FTX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JRYX654WKYFN6QPCC2RIXX4FTX/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":"2026-06-09T04:55:48Z","links":{"resolver":"https://pith.science/pith/JRYX654WKYFN6QPCC2RIXX4FTX","bundle":"https://pith.science/pith/JRYX654WKYFN6QPCC2RIXX4FTX/bundle.json","state":"https://pith.science/pith/JRYX654WKYFN6QPCC2RIXX4FTX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JRYX654WKYFN6QPCC2RIXX4FTX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JRYX654WKYFN6QPCC2RIXX4FTX","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":"68c6edab3028b78aa9cfb659f138a3ecd1e6096b399fea957c6cba212556bb4d","cross_cats_sorted":["cs.DS","cs.IT","cs.NA","math.IT","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-07T17:46:02Z","title_canon_sha256":"88214327cf8b0cafeb017580ad05962c1da5fdecca8b19b9140d730578bdb27e"},"schema_version":"1.0","source":{"id":"1501.01571","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.01571","created_at":"2026-05-18T02:29:53Z"},{"alias_kind":"arxiv_version","alias_value":"1501.01571v1","created_at":"2026-05-18T02:29:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.01571","created_at":"2026-05-18T02:29:53Z"},{"alias_kind":"pith_short_12","alias_value":"JRYX654WKYFN","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JRYX654WKYFN6QPC","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JRYX654W","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:d26dcdbc57411390e598fb01c38e7c74c3adb5b3d56f9046fca4bf9a7cc30d43","target":"graph","created_at":"2026-05-18T02:29:53Z","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":"In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point where we can conquer many (formerly) challenging problems with a page or two of arithmetic. The aim of this monograph is to describe the most successful methods from this area along with some interesting examples that these techniques can illuminate.","authors_text":"Joel A. Tropp","cross_cats":["cs.DS","cs.IT","cs.NA","math.IT","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-07T17:46:02Z","title":"An Introduction to Matrix Concentration Inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01571","kind":"arxiv","version":1},"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":"sha256:93cc0518abda4e7bbdf277f7d10ab907b08ad043b4e21ceb1f15c22020ff792e","target":"record","created_at":"2026-05-18T02:29:53Z","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":"68c6edab3028b78aa9cfb659f138a3ecd1e6096b399fea957c6cba212556bb4d","cross_cats_sorted":["cs.DS","cs.IT","cs.NA","math.IT","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-07T17:46:02Z","title_canon_sha256":"88214327cf8b0cafeb017580ad05962c1da5fdecca8b19b9140d730578bdb27e"},"schema_version":"1.0","source":{"id":"1501.01571","kind":"arxiv","version":1}},"canonical_sha256":"4c717f7796560adf41e216a28bdf859dc0983bb30c0f60058cc5a4a745df48a2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4c717f7796560adf41e216a28bdf859dc0983bb30c0f60058cc5a4a745df48a2","first_computed_at":"2026-05-18T02:29:53.744853Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:29:53.744853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HqM09t3/m6sCGvm9plEOgYQXyAvtNhqZfLiMdpw1uA2YyxuLYrG2Lkkv4S3InJr1a0PZp8aCVzwtPHjW+Ir4CA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:29:53.745311Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.01571","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93cc0518abda4e7bbdf277f7d10ab907b08ad043b4e21ceb1f15c22020ff792e","sha256:d26dcdbc57411390e598fb01c38e7c74c3adb5b3d56f9046fca4bf9a7cc30d43"],"state_sha256":"ca976ab3bcc6dd34ffd2106797e42df202dfa0458792c18fb795bf45326648cb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c1yMCd6h0/d4D6rXz8fVzAUd5ONUGQknkqcnt8oZPCWDUch7dOQNm27YLaRKq8jTSba/ROshpobPxgRKxNjvDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T04:55:48.942914Z","bundle_sha256":"4db7c7bb0186ed039284cfcf06dd78ad41ae3e745fba1e1f971bdde3106845bf"}}