{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:7O3N4KUTR7NFYGPLWNOBWUIROB","short_pith_number":"pith:7O3N4KUT","schema_version":"1.0","canonical_sha256":"fbb6de2a938fda5c19ebb35c1b511170607fb32527811d50976c2f065d746e84","source":{"kind":"arxiv","id":"1707.07953","version":1},"attestation_state":"computed","paper":{"title":"Algorithms for Positive Semidefinite Factorization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.OC","authors_text":"Arnaud Vandaele, Fran\\c{c}ois Glineur, Nicolas Gillis","submitted_at":"2017-07-25T12:32:57Z","abstract_excerpt":"This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an $m$-by-$n$ nonnegative matrix $X$ and an integer $k$, the PSD factorization problem consists in finding, if possible, symmetric $k$-by-$k$ positive semidefinite matrices $\\{A^1,...,A^m\\}$ and $\\{B^1,...,B^n\\}$ such that $X_{i,j}=\\text{trace}(A^iB^j)$ for $i=1,...,m$, and $j=1,...n$. PSD factorization is NP-hard. In this work, we introduce several local optimization schemes to tackle this problem: a fast projected gradient method and t"},"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":"1707.07953","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2017-07-25T12:32:57Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"6223b19522cfd968d93953394d8f7dd346d3915ad0230f759a613e0c9e823242","abstract_canon_sha256":"3dc8dc1c2c9390adfb42070166048752396e82fce90ecaa6433333a660078a83"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:12.062381Z","signature_b64":"TK4n4V3ajNMVaX6KbeGjMnWtRR8pMeSBgk7FWSrtd0PBhGsqvByrAts4wCG9QLxm4PJydeDI7fM58b1pgvHeCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbb6de2a938fda5c19ebb35c1b511170607fb32527811d50976c2f065d746e84","last_reissued_at":"2026-05-18T00:07:12.061715Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:12.061715Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Algorithms for Positive Semidefinite Factorization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.OC","authors_text":"Arnaud Vandaele, Fran\\c{c}ois Glineur, Nicolas Gillis","submitted_at":"2017-07-25T12:32:57Z","abstract_excerpt":"This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an $m$-by-$n$ nonnegative matrix $X$ and an integer $k$, the PSD factorization problem consists in finding, if possible, symmetric $k$-by-$k$ positive semidefinite matrices $\\{A^1,...,A^m\\}$ and $\\{B^1,...,B^n\\}$ such that $X_{i,j}=\\text{trace}(A^iB^j)$ for $i=1,...,m$, and $j=1,...n$. PSD factorization is NP-hard. In this work, we introduce several local optimization schemes to tackle this problem: a fast projected gradient method and t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.07953","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1707.07953","created_at":"2026-05-18T00:07:12.061819+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.07953v1","created_at":"2026-05-18T00:07:12.061819+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.07953","created_at":"2026-05-18T00:07:12.061819+00:00"},{"alias_kind":"pith_short_12","alias_value":"7O3N4KUTR7NF","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"7O3N4KUTR7NFYGPL","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"7O3N4KUT","created_at":"2026-05-18T12:31:05.417338+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/7O3N4KUTR7NFYGPLWNOBWUIROB","json":"https://pith.science/pith/7O3N4KUTR7NFYGPLWNOBWUIROB.json","graph_json":"https://pith.science/api/pith-number/7O3N4KUTR7NFYGPLWNOBWUIROB/graph.json","events_json":"https://pith.science/api/pith-number/7O3N4KUTR7NFYGPLWNOBWUIROB/events.json","paper":"https://pith.science/paper/7O3N4KUT"},"agent_actions":{"view_html":"https://pith.science/pith/7O3N4KUTR7NFYGPLWNOBWUIROB","download_json":"https://pith.science/pith/7O3N4KUTR7NFYGPLWNOBWUIROB.json","view_paper":"https://pith.science/paper/7O3N4KUT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.07953&json=true","fetch_graph":"https://pith.science/api/pith-number/7O3N4KUTR7NFYGPLWNOBWUIROB/graph.json","fetch_events":"https://pith.science/api/pith-number/7O3N4KUTR7NFYGPLWNOBWUIROB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7O3N4KUTR7NFYGPLWNOBWUIROB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7O3N4KUTR7NFYGPLWNOBWUIROB/action/storage_attestation","attest_author":"https://pith.science/pith/7O3N4KUTR7NFYGPLWNOBWUIROB/action/author_attestation","sign_citation":"https://pith.science/pith/7O3N4KUTR7NFYGPLWNOBWUIROB/action/citation_signature","submit_replication":"https://pith.science/pith/7O3N4KUTR7NFYGPLWNOBWUIROB/action/replication_record"}},"created_at":"2026-05-18T00:07:12.061819+00:00","updated_at":"2026-05-18T00:07:12.061819+00:00"}