{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:PR4ONNGFOE5JJB572UM5XLUBJF","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":"0b9d46d8b0e8a8644ba31d9387ef7c0d5f26ce1691630b8144806a8bd22c17da","cross_cats_sorted":["cs.IT","math.IT","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-09-04T15:47:43Z","title_canon_sha256":"ecb88a34971a7d75d5a2a0b378ad058f394517231621ef59ad288ef3bf5dbe9e"},"schema_version":"1.0","source":{"id":"1609.00951","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.00951","created_at":"2026-05-18T00:42:51Z"},{"alias_kind":"arxiv_version","alias_value":"1609.00951v3","created_at":"2026-05-18T00:42:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.00951","created_at":"2026-05-18T00:42:51Z"},{"alias_kind":"pith_short_12","alias_value":"PR4ONNGFOE5J","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PR4ONNGFOE5JJB57","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PR4ONNGF","created_at":"2026-05-18T12:30:39Z"}],"graph_snapshots":[{"event_id":"sha256:431ce4f5b3992f2b108d3ee5e6911e1981065392493b935b7f9af393ec517780","target":"graph","created_at":"2026-05-18T00:42:51Z","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 multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical convergence guarantees have only been derived for a few special divergences without regularization. In this work, we provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and regularizers. Our main result shows the sequence of iterates (i.e., pairs of basis and c","authors_text":"Renbo Zhao, Vincent Y. F. Tan","cross_cats":["cs.IT","math.IT","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-09-04T15:47:43Z","title":"A Unified Convergence Analysis of the Multiplicative Update Algorithm for Regularized Nonnegative Matrix Factorization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00951","kind":"arxiv","version":3},"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:df29e61d74970dfcd6ac0af7cf4e07f6dde2720e734885680b9dc7858437a603","target":"record","created_at":"2026-05-18T00:42:51Z","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":"0b9d46d8b0e8a8644ba31d9387ef7c0d5f26ce1691630b8144806a8bd22c17da","cross_cats_sorted":["cs.IT","math.IT","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-09-04T15:47:43Z","title_canon_sha256":"ecb88a34971a7d75d5a2a0b378ad058f394517231621ef59ad288ef3bf5dbe9e"},"schema_version":"1.0","source":{"id":"1609.00951","kind":"arxiv","version":3}},"canonical_sha256":"7c78e6b4c5713a9487bfd519dbae81495d3d5dd9becd020a64b220014b62bccb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7c78e6b4c5713a9487bfd519dbae81495d3d5dd9becd020a64b220014b62bccb","first_computed_at":"2026-05-18T00:42:51.594194Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:51.594194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S/f0r9x4yYATnUAVoU0HXKvPGRlUdpV4yqaUWrb/6h+8pbgemhLu0r7JbNnI9nFGRjymtKzTsZD8qLe9yotSCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:51.594931Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.00951","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:df29e61d74970dfcd6ac0af7cf4e07f6dde2720e734885680b9dc7858437a603","sha256:431ce4f5b3992f2b108d3ee5e6911e1981065392493b935b7f9af393ec517780"],"state_sha256":"5a1051d501258eff04330607cb05948779252b726fab3aab7f33706ece2f6113"}