{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:AKCOKL6MLDDKXELPAGWOPVTWOF","short_pith_number":"pith:AKCOKL6M","schema_version":"1.0","canonical_sha256":"0284e52fcc58c6ab916f01ace7d676715d487e3b9d2eb3e5f9fdc1c95d9f03eb","source":{"kind":"arxiv","id":"1808.05550","version":2},"attestation_state":"computed","paper":{"title":"A generalized Lieb's theorem and its applications to spectrum estimates for a sum of random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT","math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"De Huang","submitted_at":"2018-08-10T02:57:49Z","abstract_excerpt":"In this paper we prove the concavity of the $k$-trace functions, $A\\mapsto (\\text{Tr}_k[\\exp(H+\\ln A)])^{1/k}$, on the convex cone of all positive definite matrices. $\\text{Tr}_k[A]$ denotes the $k_{\\mathrm{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. As an application, we use the concavity of these $k$-trace functions to derive tail bounds and expectation estimates on the sum of the $k$ largest (or smallest) eigenvalues of a sum of random matrices."},"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":"1808.05550","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-08-10T02:57:49Z","cross_cats_sorted":["cs.IT","math.IT","math.PR","stat.TH"],"title_canon_sha256":"75d2bb528c4cfe967fe7526669e575d11309b3ded8a182de5cd8eac73dcf3b43","abstract_canon_sha256":"17056e44494840ff2b0d377eeb140deb437ad4ffe0fd67fcc2be6a8c18bb9f66"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:32.281665Z","signature_b64":"uQ75sM2EKzQ0gkTHVPmXdrVjUEW6cPlNJDrGXObAsyLt6+dzfByOo/A/YQpHGKOnHC6h/47mbMPndF3kf3CMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0284e52fcc58c6ab916f01ace7d676715d487e3b9d2eb3e5f9fdc1c95d9f03eb","last_reissued_at":"2026-05-17T23:59:32.281002Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:32.281002Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A generalized Lieb's theorem and its applications to spectrum estimates for a sum of random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT","math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"De Huang","submitted_at":"2018-08-10T02:57:49Z","abstract_excerpt":"In this paper we prove the concavity of the $k$-trace functions, $A\\mapsto (\\text{Tr}_k[\\exp(H+\\ln A)])^{1/k}$, on the convex cone of all positive definite matrices. $\\text{Tr}_k[A]$ denotes the $k_{\\mathrm{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. As an application, we use the concavity of these $k$-trace functions to derive tail bounds and expectation estimates on the sum of the $k$ largest (or smallest) eigenvalues of a sum of random matrices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.05550","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":""},"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":"1808.05550","created_at":"2026-05-17T23:59:32.281100+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.05550v2","created_at":"2026-05-17T23:59:32.281100+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.05550","created_at":"2026-05-17T23:59:32.281100+00:00"},{"alias_kind":"pith_short_12","alias_value":"AKCOKL6MLDDK","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_16","alias_value":"AKCOKL6MLDDKXELP","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_8","alias_value":"AKCOKL6M","created_at":"2026-05-18T12:32:13.499390+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AKCOKL6MLDDKXELPAGWOPVTWOF","json":"https://pith.science/pith/AKCOKL6MLDDKXELPAGWOPVTWOF.json","graph_json":"https://pith.science/api/pith-number/AKCOKL6MLDDKXELPAGWOPVTWOF/graph.json","events_json":"https://pith.science/api/pith-number/AKCOKL6MLDDKXELPAGWOPVTWOF/events.json","paper":"https://pith.science/paper/AKCOKL6M"},"agent_actions":{"view_html":"https://pith.science/pith/AKCOKL6MLDDKXELPAGWOPVTWOF","download_json":"https://pith.science/pith/AKCOKL6MLDDKXELPAGWOPVTWOF.json","view_paper":"https://pith.science/paper/AKCOKL6M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.05550&json=true","fetch_graph":"https://pith.science/api/pith-number/AKCOKL6MLDDKXELPAGWOPVTWOF/graph.json","fetch_events":"https://pith.science/api/pith-number/AKCOKL6MLDDKXELPAGWOPVTWOF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AKCOKL6MLDDKXELPAGWOPVTWOF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AKCOKL6MLDDKXELPAGWOPVTWOF/action/storage_attestation","attest_author":"https://pith.science/pith/AKCOKL6MLDDKXELPAGWOPVTWOF/action/author_attestation","sign_citation":"https://pith.science/pith/AKCOKL6MLDDKXELPAGWOPVTWOF/action/citation_signature","submit_replication":"https://pith.science/pith/AKCOKL6MLDDKXELPAGWOPVTWOF/action/replication_record"}},"created_at":"2026-05-17T23:59:32.281100+00:00","updated_at":"2026-05-17T23:59:32.281100+00:00"}