{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:MP6DAFF3FAWDR77RUXIS7CVCY2","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":"89582f6b44a225ccc311d40fa73fd44a0adf04e393c0966166ce8e8c0b80e543","cross_cats_sorted":["cs.DM","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-18T15:16:12Z","title_canon_sha256":"d7537c9dd5adef7028d9f3f8b389bba5a591f93bc587c8a2e246980d5131fe64"},"schema_version":"1.0","source":{"id":"1203.3961","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.3961","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"arxiv_version","alias_value":"1203.3961v4","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.3961","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"pith_short_12","alias_value":"MP6DAFF3FAWD","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MP6DAFF3FAWDR77R","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MP6DAFF3","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:e10f66f387fdb763fae79f5cbf487641731d3c85625d3743490659ea4a3404c0","target":"graph","created_at":"2026-05-18T03:07:02Z","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":"The positive semidefinite rank of a nonnegative $(m\\times n)$-matrix~$S$ is the minimum number~$q$ such that there exist positive semidefinite $(q\\times q)$-matrices $A_1,\\dots,A_m$, $B_1,\\dots,B_n$ such that $S(k,\\ell) = \\mbox{tr}(A_k^* B_\\ell)$.\n  The most important, lower bound technique for nonnegative rank is solely based on the support of the matrix S, i.e., its zero/non-zero pattern. In this paper, we characterize the power of lower bounds on positive semidefinite rank based on solely on the support.","authors_text":"Dirk Oliver Theis, Troy Lee","cross_cats":["cs.DM","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-18T15:16:12Z","title":"Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.3961","kind":"arxiv","version":4},"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:fef8f58c15909bb39839f50816a43d575f0415974ca471f19195ae93bb28e27b","target":"record","created_at":"2026-05-18T03:07:02Z","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":"89582f6b44a225ccc311d40fa73fd44a0adf04e393c0966166ce8e8c0b80e543","cross_cats_sorted":["cs.DM","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-18T15:16:12Z","title_canon_sha256":"d7537c9dd5adef7028d9f3f8b389bba5a591f93bc587c8a2e246980d5131fe64"},"schema_version":"1.0","source":{"id":"1203.3961","kind":"arxiv","version":4}},"canonical_sha256":"63fc3014bb282c38fff1a5d12f8aa2c680c36ca91457a861ae7e486e1aec88f0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63fc3014bb282c38fff1a5d12f8aa2c680c36ca91457a861ae7e486e1aec88f0","first_computed_at":"2026-05-18T03:07:02.274119Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:02.274119Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U4saxO65zOWIJkUKNmin+ghD58BswYZY1d3hCn7gQauIZMPPpOZ0c9AIwStGFhCpB4ibZpR8uMEdsanqiuuCCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:02.274688Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.3961","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fef8f58c15909bb39839f50816a43d575f0415974ca471f19195ae93bb28e27b","sha256:e10f66f387fdb763fae79f5cbf487641731d3c85625d3743490659ea4a3404c0"],"state_sha256":"52df06f64ae52195d7db214ac64b059abd98740a5199b35ece0b10441ced0ca7"}