{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:MP6DAFF3FAWDR77RUXIS7CVCY2","short_pith_number":"pith:MP6DAFF3","schema_version":"1.0","canonical_sha256":"63fc3014bb282c38fff1a5d12f8aa2c680c36ca91457a861ae7e486e1aec88f0","source":{"kind":"arxiv","id":"1203.3961","version":4},"attestation_state":"computed","paper":{"title":"Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.OC"],"primary_cat":"math.CO","authors_text":"Dirk Oliver Theis, Troy Lee","submitted_at":"2012-03-18T15:16:12Z","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."},"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":"1203.3961","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-18T15:16:12Z","cross_cats_sorted":["cs.DM","math.OC"],"title_canon_sha256":"d7537c9dd5adef7028d9f3f8b389bba5a591f93bc587c8a2e246980d5131fe64","abstract_canon_sha256":"89582f6b44a225ccc311d40fa73fd44a0adf04e393c0966166ce8e8c0b80e543"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:02.274688Z","signature_b64":"U4saxO65zOWIJkUKNmin+ghD58BswYZY1d3hCn7gQauIZMPPpOZ0c9AIwStGFhCpB4ibZpR8uMEdsanqiuuCCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63fc3014bb282c38fff1a5d12f8aa2c680c36ca91457a861ae7e486e1aec88f0","last_reissued_at":"2026-05-18T03:07:02.274119Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:02.274119Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.OC"],"primary_cat":"math.CO","authors_text":"Dirk Oliver Theis, Troy Lee","submitted_at":"2012-03-18T15:16:12Z","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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.3961","kind":"arxiv","version":4},"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":"1203.3961","created_at":"2026-05-18T03:07:02.274182+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.3961v4","created_at":"2026-05-18T03:07:02.274182+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.3961","created_at":"2026-05-18T03:07:02.274182+00:00"},{"alias_kind":"pith_short_12","alias_value":"MP6DAFF3FAWD","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"MP6DAFF3FAWDR77R","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"MP6DAFF3","created_at":"2026-05-18T12:27:14.488303+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/MP6DAFF3FAWDR77RUXIS7CVCY2","json":"https://pith.science/pith/MP6DAFF3FAWDR77RUXIS7CVCY2.json","graph_json":"https://pith.science/api/pith-number/MP6DAFF3FAWDR77RUXIS7CVCY2/graph.json","events_json":"https://pith.science/api/pith-number/MP6DAFF3FAWDR77RUXIS7CVCY2/events.json","paper":"https://pith.science/paper/MP6DAFF3"},"agent_actions":{"view_html":"https://pith.science/pith/MP6DAFF3FAWDR77RUXIS7CVCY2","download_json":"https://pith.science/pith/MP6DAFF3FAWDR77RUXIS7CVCY2.json","view_paper":"https://pith.science/paper/MP6DAFF3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.3961&json=true","fetch_graph":"https://pith.science/api/pith-number/MP6DAFF3FAWDR77RUXIS7CVCY2/graph.json","fetch_events":"https://pith.science/api/pith-number/MP6DAFF3FAWDR77RUXIS7CVCY2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MP6DAFF3FAWDR77RUXIS7CVCY2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MP6DAFF3FAWDR77RUXIS7CVCY2/action/storage_attestation","attest_author":"https://pith.science/pith/MP6DAFF3FAWDR77RUXIS7CVCY2/action/author_attestation","sign_citation":"https://pith.science/pith/MP6DAFF3FAWDR77RUXIS7CVCY2/action/citation_signature","submit_replication":"https://pith.science/pith/MP6DAFF3FAWDR77RUXIS7CVCY2/action/replication_record"}},"created_at":"2026-05-18T03:07:02.274182+00:00","updated_at":"2026-05-18T03:07:02.274182+00:00"}