{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:QXL6WBYI5EOWBNGXLYYY2IPUQM","short_pith_number":"pith:QXL6WBYI","canonical_record":{"source":{"id":"1512.07548","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2015-12-23T17:12:06Z","cross_cats_sorted":[],"title_canon_sha256":"40d9d8d87366d05847de40b9946a1a8a08a41cf15cf44c5eb3cc44980919521e","abstract_canon_sha256":"5a0a540e49aab8d5e26a23fb0c38304b6123fb63a42b9605822d47e362d08d28"},"schema_version":"1.0"},"canonical_sha256":"85d7eb0708e91d60b4d75e318d21f4832121cd880f05a5344c89bf65a1ef04d0","source":{"kind":"arxiv","id":"1512.07548","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.07548","created_at":"2026-05-18T01:23:49Z"},{"alias_kind":"arxiv_version","alias_value":"1512.07548v1","created_at":"2026-05-18T01:23:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.07548","created_at":"2026-05-18T01:23:49Z"},{"alias_kind":"pith_short_12","alias_value":"QXL6WBYI5EOW","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"QXL6WBYI5EOWBNGX","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"QXL6WBYI","created_at":"2026-05-18T12:29:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:QXL6WBYI5EOWBNGXLYYY2IPUQM","target":"record","payload":{"canonical_record":{"source":{"id":"1512.07548","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2015-12-23T17:12:06Z","cross_cats_sorted":[],"title_canon_sha256":"40d9d8d87366d05847de40b9946a1a8a08a41cf15cf44c5eb3cc44980919521e","abstract_canon_sha256":"5a0a540e49aab8d5e26a23fb0c38304b6123fb63a42b9605822d47e362d08d28"},"schema_version":"1.0"},"canonical_sha256":"85d7eb0708e91d60b4d75e318d21f4832121cd880f05a5344c89bf65a1ef04d0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:49.415305Z","signature_b64":"dNKBGH8TU5Gc48MTG911wcbNpjuSVJm8eF+DV5wBNT5Y0m4Mdrfx1GwPNKnmNjQizGuCs1o1sACPbOespa9BCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85d7eb0708e91d60b4d75e318d21f4832121cd880f05a5344c89bf65a1ef04d0","last_reissued_at":"2026-05-18T01:23:49.414626Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:49.414626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.07548","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:23:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l3OGJuo697qv3x/KWmX9YtjrvcLDkncxn7Iia+MvhCC7ov/R5++4U/lcIKmWu9KEqwVvS0ngzpdM6gvKu3svBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T16:38:01.518173Z"},"content_sha256":"c3d3e20006964203ed1ea94d83afca1c128d921e0739ce64e66aaffcbd627861","schema_version":"1.0","event_id":"sha256:c3d3e20006964203ed1ea94d83afca1c128d921e0739ce64e66aaffcbd627861"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:QXL6WBYI5EOWBNGXLYYY2IPUQM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"k-Means Clustering Is Matrix Factorization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"stat.ML","authors_text":"Christian Bauckhage","submitted_at":"2015-12-23T17:12:06Z","abstract_excerpt":"We show that the objective function of conventional k-means clustering can be expressed as the Frobenius norm of the difference of a data matrix and a low rank approximation of that data matrix. In short, we show that k-means clustering is a matrix factorization problem. These notes are meant as a reference and intended to provide a guided tour towards a result that is often mentioned but seldom made explicit in the literature."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07548","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:23:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w14rd8KLYYF3tJG8RPn+Im7WnmAu+yBt3RbVWv0mcL2GyDr/y8tymPeUWLBuHMSZgAmVRWyH5N5qPLNQ5Ig+Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T16:38:01.518798Z"},"content_sha256":"ddad3e96ed3b11ee758a03db688cdb51c26844936d047d65bd74a2c4a7bf9528","schema_version":"1.0","event_id":"sha256:ddad3e96ed3b11ee758a03db688cdb51c26844936d047d65bd74a2c4a7bf9528"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QXL6WBYI5EOWBNGXLYYY2IPUQM/bundle.json","state_url":"https://pith.science/pith/QXL6WBYI5EOWBNGXLYYY2IPUQM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QXL6WBYI5EOWBNGXLYYY2IPUQM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T16:38:01Z","links":{"resolver":"https://pith.science/pith/QXL6WBYI5EOWBNGXLYYY2IPUQM","bundle":"https://pith.science/pith/QXL6WBYI5EOWBNGXLYYY2IPUQM/bundle.json","state":"https://pith.science/pith/QXL6WBYI5EOWBNGXLYYY2IPUQM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QXL6WBYI5EOWBNGXLYYY2IPUQM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QXL6WBYI5EOWBNGXLYYY2IPUQM","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":"5a0a540e49aab8d5e26a23fb0c38304b6123fb63a42b9605822d47e362d08d28","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2015-12-23T17:12:06Z","title_canon_sha256":"40d9d8d87366d05847de40b9946a1a8a08a41cf15cf44c5eb3cc44980919521e"},"schema_version":"1.0","source":{"id":"1512.07548","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.07548","created_at":"2026-05-18T01:23:49Z"},{"alias_kind":"arxiv_version","alias_value":"1512.07548v1","created_at":"2026-05-18T01:23:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.07548","created_at":"2026-05-18T01:23:49Z"},{"alias_kind":"pith_short_12","alias_value":"QXL6WBYI5EOW","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"QXL6WBYI5EOWBNGX","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"QXL6WBYI","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:ddad3e96ed3b11ee758a03db688cdb51c26844936d047d65bd74a2c4a7bf9528","target":"graph","created_at":"2026-05-18T01:23:49Z","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":"We show that the objective function of conventional k-means clustering can be expressed as the Frobenius norm of the difference of a data matrix and a low rank approximation of that data matrix. In short, we show that k-means clustering is a matrix factorization problem. These notes are meant as a reference and intended to provide a guided tour towards a result that is often mentioned but seldom made explicit in the literature.","authors_text":"Christian Bauckhage","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2015-12-23T17:12:06Z","title":"k-Means Clustering Is Matrix Factorization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07548","kind":"arxiv","version":1},"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:c3d3e20006964203ed1ea94d83afca1c128d921e0739ce64e66aaffcbd627861","target":"record","created_at":"2026-05-18T01:23:49Z","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":"5a0a540e49aab8d5e26a23fb0c38304b6123fb63a42b9605822d47e362d08d28","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2015-12-23T17:12:06Z","title_canon_sha256":"40d9d8d87366d05847de40b9946a1a8a08a41cf15cf44c5eb3cc44980919521e"},"schema_version":"1.0","source":{"id":"1512.07548","kind":"arxiv","version":1}},"canonical_sha256":"85d7eb0708e91d60b4d75e318d21f4832121cd880f05a5344c89bf65a1ef04d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"85d7eb0708e91d60b4d75e318d21f4832121cd880f05a5344c89bf65a1ef04d0","first_computed_at":"2026-05-18T01:23:49.414626Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:23:49.414626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dNKBGH8TU5Gc48MTG911wcbNpjuSVJm8eF+DV5wBNT5Y0m4Mdrfx1GwPNKnmNjQizGuCs1o1sACPbOespa9BCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:23:49.415305Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.07548","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3d3e20006964203ed1ea94d83afca1c128d921e0739ce64e66aaffcbd627861","sha256:ddad3e96ed3b11ee758a03db688cdb51c26844936d047d65bd74a2c4a7bf9528"],"state_sha256":"416e2f4dee510fc97c785cac3405ff7ac6016192c95bc998694326b7f4331801"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A/WSJCqnq5R67X1TcnDukkTaGwE+3+53wUKxKfRBevLHgTTCeAUvb4M9P4iqtHTWx7Tl4IjRaFRenPfQM2JEDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T16:38:01.521960Z","bundle_sha256":"f1082b68c57534b3759d54dbdf25fe5e2ff9c879041c0319cb926c1be4a2266f"}}