{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:TIWDDJKPHXDLUPM3U6PENMEI2P","short_pith_number":"pith:TIWDDJKP","canonical_record":{"source":{"id":"1612.07516","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2016-12-22T10:10:11Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"d5959bdec9695d0efc09a308779ccc6a323b37fe757f623223fb9a46ff36baa3","abstract_canon_sha256":"e8a6658bc9c14660e2a06703547e078584a8dbe45b99096b06f8bcc54742f510"},"schema_version":"1.0"},"canonical_sha256":"9a2c31a54f3dc6ba3d9ba79e46b088d3ffe17583affefcc7e6e569dd77ddb9a2","source":{"kind":"arxiv","id":"1612.07516","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.07516","created_at":"2026-05-18T00:04:41Z"},{"alias_kind":"arxiv_version","alias_value":"1612.07516v3","created_at":"2026-05-18T00:04:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.07516","created_at":"2026-05-18T00:04:41Z"},{"alias_kind":"pith_short_12","alias_value":"TIWDDJKPHXDL","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"TIWDDJKPHXDLUPM3","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"TIWDDJKP","created_at":"2026-05-18T12:30:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:TIWDDJKPHXDLUPM3U6PENMEI2P","target":"record","payload":{"canonical_record":{"source":{"id":"1612.07516","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2016-12-22T10:10:11Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"d5959bdec9695d0efc09a308779ccc6a323b37fe757f623223fb9a46ff36baa3","abstract_canon_sha256":"e8a6658bc9c14660e2a06703547e078584a8dbe45b99096b06f8bcc54742f510"},"schema_version":"1.0"},"canonical_sha256":"9a2c31a54f3dc6ba3d9ba79e46b088d3ffe17583affefcc7e6e569dd77ddb9a2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:41.936879Z","signature_b64":"8QjTzpDCAOvLYj1kq1utr/kgwWvLlxpaAkrJyBS5YP09T8AA1/kghDAY24i039PExfhTxwKo/OPgeYbXkySKDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a2c31a54f3dc6ba3d9ba79e46b088d3ffe17583affefcc7e6e569dd77ddb9a2","last_reissued_at":"2026-05-18T00:04:41.936112Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:41.936112Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.07516","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:04:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ADsfC9K7hIQL1BZod9R3bvLJGje1nOSBMAmbfUFz+mgpLo3SoKLdWKOEAsT/htQ6kzGYzvVOI7P6Op6vsNXgBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T16:39:05.923826Z"},"content_sha256":"f4c52d269d820ef8f877291b8f79c7dfab5b16955741d6afe10252caefdb5459","schema_version":"1.0","event_id":"sha256:f4c52d269d820ef8f877291b8f79c7dfab5b16955741d6afe10252caefdb5459"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:TIWDDJKPHXDLUPM3U6PENMEI2P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Coreset Constructions for the Fuzzy $K$-Means Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.LG","authors_text":"Johannes Bl\\\"omer, Kathrin Bujna, Sascha Brauer","submitted_at":"2016-12-22T10:10:11Z","abstract_excerpt":"The fuzzy $K$-means problem is a popular generalization of the well-known $K$-means problem to soft clusterings. We present the first coresets for fuzzy $K$-means with size linear in the dimension, polynomial in the number of clusters, and poly-logarithmic in the number of points. We show that these coresets can be employed in the computation of a $(1+\\epsilon)$-approximation for fuzzy $K$-means, improving previously presented results. We further show that our coresets can be maintained in an insertion-only streaming setting, where data points arrive one-by-one."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07516","kind":"arxiv","version":3},"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":"2026-05-18T00:04:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Txi+18BDySaDaGANtPKei7CCznSxZVBZGfYSGgtyyFCVnnTvrJn3DWwTD0CAQnVd0V4GKxvLEAjoLJupA8qlCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T16:39:05.924268Z"},"content_sha256":"522fcfcf765510e7af36742787502f8477bea0933aae5590369e5d817f92d249","schema_version":"1.0","event_id":"sha256:522fcfcf765510e7af36742787502f8477bea0933aae5590369e5d817f92d249"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TIWDDJKPHXDLUPM3U6PENMEI2P/bundle.json","state_url":"https://pith.science/pith/TIWDDJKPHXDLUPM3U6PENMEI2P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TIWDDJKPHXDLUPM3U6PENMEI2P/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":"2026-08-10T16:39:05Z","links":{"resolver":"https://pith.science/pith/TIWDDJKPHXDLUPM3U6PENMEI2P","bundle":"https://pith.science/pith/TIWDDJKPHXDLUPM3U6PENMEI2P/bundle.json","state":"https://pith.science/pith/TIWDDJKPHXDLUPM3U6PENMEI2P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TIWDDJKPHXDLUPM3U6PENMEI2P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:TIWDDJKPHXDLUPM3U6PENMEI2P","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":"e8a6658bc9c14660e2a06703547e078584a8dbe45b99096b06f8bcc54742f510","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2016-12-22T10:10:11Z","title_canon_sha256":"d5959bdec9695d0efc09a308779ccc6a323b37fe757f623223fb9a46ff36baa3"},"schema_version":"1.0","source":{"id":"1612.07516","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.07516","created_at":"2026-05-18T00:04:41Z"},{"alias_kind":"arxiv_version","alias_value":"1612.07516v3","created_at":"2026-05-18T00:04:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.07516","created_at":"2026-05-18T00:04:41Z"},{"alias_kind":"pith_short_12","alias_value":"TIWDDJKPHXDL","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"TIWDDJKPHXDLUPM3","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"TIWDDJKP","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:522fcfcf765510e7af36742787502f8477bea0933aae5590369e5d817f92d249","target":"graph","created_at":"2026-05-18T00:04:41Z","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 fuzzy $K$-means problem is a popular generalization of the well-known $K$-means problem to soft clusterings. We present the first coresets for fuzzy $K$-means with size linear in the dimension, polynomial in the number of clusters, and poly-logarithmic in the number of points. We show that these coresets can be employed in the computation of a $(1+\\epsilon)$-approximation for fuzzy $K$-means, improving previously presented results. We further show that our coresets can be maintained in an insertion-only streaming setting, where data points arrive one-by-one.","authors_text":"Johannes Bl\\\"omer, Kathrin Bujna, Sascha Brauer","cross_cats":["cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2016-12-22T10:10:11Z","title":"On Coreset Constructions for the Fuzzy $K$-Means Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07516","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:f4c52d269d820ef8f877291b8f79c7dfab5b16955741d6afe10252caefdb5459","target":"record","created_at":"2026-05-18T00:04:41Z","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":"e8a6658bc9c14660e2a06703547e078584a8dbe45b99096b06f8bcc54742f510","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2016-12-22T10:10:11Z","title_canon_sha256":"d5959bdec9695d0efc09a308779ccc6a323b37fe757f623223fb9a46ff36baa3"},"schema_version":"1.0","source":{"id":"1612.07516","kind":"arxiv","version":3}},"canonical_sha256":"9a2c31a54f3dc6ba3d9ba79e46b088d3ffe17583affefcc7e6e569dd77ddb9a2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a2c31a54f3dc6ba3d9ba79e46b088d3ffe17583affefcc7e6e569dd77ddb9a2","first_computed_at":"2026-05-18T00:04:41.936112Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:04:41.936112Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8QjTzpDCAOvLYj1kq1utr/kgwWvLlxpaAkrJyBS5YP09T8AA1/kghDAY24i039PExfhTxwKo/OPgeYbXkySKDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:04:41.936879Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.07516","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f4c52d269d820ef8f877291b8f79c7dfab5b16955741d6afe10252caefdb5459","sha256:522fcfcf765510e7af36742787502f8477bea0933aae5590369e5d817f92d249"],"state_sha256":"e064c321afce8d9391b95ee4164ec2b8fef7cf3cbbd16e9ca9136d20c1b60b64"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1kKK6ROM+/5D+dd2jgzQzKrmQUnk04X8GM0D9hUsl0WOZ4yQWXT8XJw5zjS0T0uHEulDbRvmcaIBo3m+4cOYBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T16:39:05.928026Z","bundle_sha256":"974aaee2fe3edcc3777093ec597922821433e3a9fc718380c354890e088bca13"}}