{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:33EEZZLDDNAMXMF5S6QNZHVBFI","short_pith_number":"pith:33EEZZLD","canonical_record":{"source":{"id":"2607.01945","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2026-07-02T09:34:44Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"520237b9f2aab784888ea8281e97bfde89efe810ebae4577cb19f7e7d611b240","abstract_canon_sha256":"097c2ee8b43337d0107b1bfae45c9961557e65da2596dc445cdc8647b7928895"},"schema_version":"1.0"},"canonical_sha256":"dec84ce5631b40cbb0bd97a0dc9ea12a2f43aac50d65dd11e52c905c2f394392","source":{"kind":"arxiv","id":"2607.01945","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01945","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01945v1","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01945","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_12","alias_value":"33EEZZLDDNAM","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_16","alias_value":"33EEZZLDDNAMXMF5","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_8","alias_value":"33EEZZLD","created_at":"2026-07-03T01:17:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:33EEZZLDDNAMXMF5S6QNZHVBFI","target":"record","payload":{"canonical_record":{"source":{"id":"2607.01945","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2026-07-02T09:34:44Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"520237b9f2aab784888ea8281e97bfde89efe810ebae4577cb19f7e7d611b240","abstract_canon_sha256":"097c2ee8b43337d0107b1bfae45c9961557e65da2596dc445cdc8647b7928895"},"schema_version":"1.0"},"canonical_sha256":"dec84ce5631b40cbb0bd97a0dc9ea12a2f43aac50d65dd11e52c905c2f394392","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:34.642066Z","signature_b64":"zWxlLzRmKibKobgGh25q4Dg8iTSnO3Gz8sT9ej+4Hm44vc9SNDqNT3u+rdGLabuTaP8UoMk/EzhrWSyZaZB9DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dec84ce5631b40cbb0bd97a0dc9ea12a2f43aac50d65dd11e52c905c2f394392","last_reissued_at":"2026-07-03T01:17:34.641650Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:34.641650Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.01945","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":"2026-07-03T01:17:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CdUoJ4CeUpRTuL+tWrJO2f5DyzBINcPMjc+uj2lYFJoz9HIzUMIpkuJzddwk4TPTS2r7iURDm4rbC5+VVxeXBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T23:38:39.828864Z"},"content_sha256":"2676d81a9b4b8ad7f67a620c257e69c1097dcf5daa42a8fa4a82140f6e38574b","schema_version":"1.0","event_id":"sha256:2676d81a9b4b8ad7f67a620c257e69c1097dcf5daa42a8fa4a82140f6e38574b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:33EEZZLDDNAMXMF5S6QNZHVBFI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Statistical Properties of $k$-means Clustering for Data Missing Completely at Random","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Xin Guan","submitted_at":"2026-07-02T09:34:44Z","abstract_excerpt":"The classical $k$-means clustering cannot be directly used to incomplete data, and existing $k$-means-based clustering for missing data primarily focus on improving the practical accuracy of clustering, whereas most of them lack theoretical guarantees in the asymptotic sense. In this paper, we investigate the statistical properties of $k$-means clustering in the presence of missing data. We first establish the $\\sqrt{n}$-excess risk bound and prove the consistency of the estimated cluster centers under general missing mechanisms. For the Missing Completely at Random (MCAR) mechanism, we furthe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01945","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.01945/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-03T01:17:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ca/GTTq4hnt4Y6ztqmPvNTOVxIN6dKjUhGiWu/nXRl9nmicfKnO54csd2grWLMvCwqHjAPDdqaM6ivo3iZ0ZCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T23:38:39.829492Z"},"content_sha256":"9669788e8b9bd61b29529cc4be19561ea4e24fe535efabfdd47d770baa5dfb71","schema_version":"1.0","event_id":"sha256:9669788e8b9bd61b29529cc4be19561ea4e24fe535efabfdd47d770baa5dfb71"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/33EEZZLDDNAMXMF5S6QNZHVBFI/bundle.json","state_url":"https://pith.science/pith/33EEZZLDDNAMXMF5S6QNZHVBFI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/33EEZZLDDNAMXMF5S6QNZHVBFI/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":"2026-07-03T23:38:39Z","links":{"resolver":"https://pith.science/pith/33EEZZLDDNAMXMF5S6QNZHVBFI","bundle":"https://pith.science/pith/33EEZZLDDNAMXMF5S6QNZHVBFI/bundle.json","state":"https://pith.science/pith/33EEZZLDDNAMXMF5S6QNZHVBFI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/33EEZZLDDNAMXMF5S6QNZHVBFI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:33EEZZLDDNAMXMF5S6QNZHVBFI","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":"097c2ee8b43337d0107b1bfae45c9961557e65da2596dc445cdc8647b7928895","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2026-07-02T09:34:44Z","title_canon_sha256":"520237b9f2aab784888ea8281e97bfde89efe810ebae4577cb19f7e7d611b240"},"schema_version":"1.0","source":{"id":"2607.01945","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01945","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01945v1","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01945","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_12","alias_value":"33EEZZLDDNAM","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_16","alias_value":"33EEZZLDDNAMXMF5","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_8","alias_value":"33EEZZLD","created_at":"2026-07-03T01:17:34Z"}],"graph_snapshots":[{"event_id":"sha256:9669788e8b9bd61b29529cc4be19561ea4e24fe535efabfdd47d770baa5dfb71","target":"graph","created_at":"2026-07-03T01:17:34Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.01945/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The classical $k$-means clustering cannot be directly used to incomplete data, and existing $k$-means-based clustering for missing data primarily focus on improving the practical accuracy of clustering, whereas most of them lack theoretical guarantees in the asymptotic sense. In this paper, we investigate the statistical properties of $k$-means clustering in the presence of missing data. We first establish the $\\sqrt{n}$-excess risk bound and prove the consistency of the estimated cluster centers under general missing mechanisms. For the Missing Completely at Random (MCAR) mechanism, we furthe","authors_text":"Xin Guan","cross_cats":["cs.LG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2026-07-02T09:34:44Z","title":"Statistical Properties of $k$-means Clustering for Data Missing Completely at Random"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01945","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":"sha256:2676d81a9b4b8ad7f67a620c257e69c1097dcf5daa42a8fa4a82140f6e38574b","target":"record","created_at":"2026-07-03T01:17:34Z","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":"097c2ee8b43337d0107b1bfae45c9961557e65da2596dc445cdc8647b7928895","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2026-07-02T09:34:44Z","title_canon_sha256":"520237b9f2aab784888ea8281e97bfde89efe810ebae4577cb19f7e7d611b240"},"schema_version":"1.0","source":{"id":"2607.01945","kind":"arxiv","version":1}},"canonical_sha256":"dec84ce5631b40cbb0bd97a0dc9ea12a2f43aac50d65dd11e52c905c2f394392","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dec84ce5631b40cbb0bd97a0dc9ea12a2f43aac50d65dd11e52c905c2f394392","first_computed_at":"2026-07-03T01:17:34.641650Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:34.641650Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zWxlLzRmKibKobgGh25q4Dg8iTSnO3Gz8sT9ej+4Hm44vc9SNDqNT3u+rdGLabuTaP8UoMk/EzhrWSyZaZB9DA==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:34.642066Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.01945","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2676d81a9b4b8ad7f67a620c257e69c1097dcf5daa42a8fa4a82140f6e38574b","sha256:9669788e8b9bd61b29529cc4be19561ea4e24fe535efabfdd47d770baa5dfb71"],"state_sha256":"2a708a7731bcccc00e2c5278bc0b3e8901c51fa9c0051385122fa9d3cb03b62f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9VxfHPJ1IqZL3Ncx/gnY+AfLIFV/jEae0fwTcrsGCvhjlJbMFlFl47HS90VbpJCQFMKxXvasSimpVHzt2PRUDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T23:38:39.832890Z","bundle_sha256":"50778b6db902ff5feca93b6162dcc5bcdbec2b324667331d8757635358b43e15"}}