{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2004:OG5QTN6FLKFUSBHBXFB5RSFVX6","short_pith_number":"pith:OG5QTN6F","schema_version":"1.0","canonical_sha256":"71bb09b7c55a8b4904e1b943d8c8b5bfa88c628689c07959cb7d5eb402188698","source":{"kind":"arxiv","id":"math/0412070","version":2},"attestation_state":"computed","paper":{"title":"Nonnegative Matrix Factorization and I-Divergence Alternating Minimization","license":"","headline":"","cross_cats":["math.PR"],"primary_cat":"math.OC","authors_text":"Lorenzo Finesso, Peter Spreij","submitted_at":"2004-12-03T12:02:17Z","abstract_excerpt":"In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix $V \\in \\R_+^{m\\times n}$ find, for assigned $k$, nonnegative matrices $W\\in\\R_+^{m\\times k}$ and $H\\in\\R_+^{k\\times n}$ such that $V=WH$. Exact, non trivial, nonnegative factorizations do not always exist, hence it is interesting to pose the approximate NMF problem. The criterion which is commonly employed is I-divergence between nonnegative matrices. The problem becomes that of finding, for assigned $k$, the factorization $WH$ closest to $V$ in I-divergence. An iterative alg"},"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":"math/0412070","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.OC","submitted_at":"2004-12-03T12:02:17Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"5e3c887dd5ea8b5fa4e6dd180f9d626e94d49992bc5236ada0d5444680905678","abstract_canon_sha256":"8f6a92b64976e9fd875d5ba784892e9d20e19f6d3abc430607ae0eeb2a9a1359"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:10.314528Z","signature_b64":"WGxI2gHTsugHNbjdI1rHryMC47Uq8PNBFT5likVUL7fNH6D437PkuJ1OXHwQpe0qQlsQZDgxGete5KuHzNgXAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71bb09b7c55a8b4904e1b943d8c8b5bfa88c628689c07959cb7d5eb402188698","last_reissued_at":"2026-05-18T02:48:10.313921Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:10.313921Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nonnegative Matrix Factorization and I-Divergence Alternating Minimization","license":"","headline":"","cross_cats":["math.PR"],"primary_cat":"math.OC","authors_text":"Lorenzo Finesso, Peter Spreij","submitted_at":"2004-12-03T12:02:17Z","abstract_excerpt":"In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix $V \\in \\R_+^{m\\times n}$ find, for assigned $k$, nonnegative matrices $W\\in\\R_+^{m\\times k}$ and $H\\in\\R_+^{k\\times n}$ such that $V=WH$. Exact, non trivial, nonnegative factorizations do not always exist, hence it is interesting to pose the approximate NMF problem. The criterion which is commonly employed is I-divergence between nonnegative matrices. The problem becomes that of finding, for assigned $k$, the factorization $WH$ closest to $V$ in I-divergence. An iterative alg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0412070","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":"math/0412070","created_at":"2026-05-18T02:48:10.314003+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0412070v2","created_at":"2026-05-18T02:48:10.314003+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0412070","created_at":"2026-05-18T02:48:10.314003+00:00"},{"alias_kind":"pith_short_12","alias_value":"OG5QTN6FLKFU","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_16","alias_value":"OG5QTN6FLKFUSBHB","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_8","alias_value":"OG5QTN6F","created_at":"2026-05-18T12:25:52.687210+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/OG5QTN6FLKFUSBHBXFB5RSFVX6","json":"https://pith.science/pith/OG5QTN6FLKFUSBHBXFB5RSFVX6.json","graph_json":"https://pith.science/api/pith-number/OG5QTN6FLKFUSBHBXFB5RSFVX6/graph.json","events_json":"https://pith.science/api/pith-number/OG5QTN6FLKFUSBHBXFB5RSFVX6/events.json","paper":"https://pith.science/paper/OG5QTN6F"},"agent_actions":{"view_html":"https://pith.science/pith/OG5QTN6FLKFUSBHBXFB5RSFVX6","download_json":"https://pith.science/pith/OG5QTN6FLKFUSBHBXFB5RSFVX6.json","view_paper":"https://pith.science/paper/OG5QTN6F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0412070&json=true","fetch_graph":"https://pith.science/api/pith-number/OG5QTN6FLKFUSBHBXFB5RSFVX6/graph.json","fetch_events":"https://pith.science/api/pith-number/OG5QTN6FLKFUSBHBXFB5RSFVX6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OG5QTN6FLKFUSBHBXFB5RSFVX6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OG5QTN6FLKFUSBHBXFB5RSFVX6/action/storage_attestation","attest_author":"https://pith.science/pith/OG5QTN6FLKFUSBHBXFB5RSFVX6/action/author_attestation","sign_citation":"https://pith.science/pith/OG5QTN6FLKFUSBHBXFB5RSFVX6/action/citation_signature","submit_replication":"https://pith.science/pith/OG5QTN6FLKFUSBHBXFB5RSFVX6/action/replication_record"}},"created_at":"2026-05-18T02:48:10.314003+00:00","updated_at":"2026-05-18T02:48:10.314003+00:00"}