{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:NDQO6ACKZ3G5C32CAYQPEHHY7Q","short_pith_number":"pith:NDQO6ACK","canonical_record":{"source":{"id":"2304.00419","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-04-02T00:58:29Z","cross_cats_sorted":["cs.AI","cs.DS"],"title_canon_sha256":"65e50a8e25bf4274c29971b92a2833b106350b33008604ac19364d106099b0c8","abstract_canon_sha256":"0f8f14a996f6ecab179ae341e32f6312e575fcbdcc595fe365742f537617663c"},"schema_version":"1.0"},"canonical_sha256":"68e0ef004acecdd16f420620f21cf8fc16339aff9ab7e8bf40f58676b80b49e8","source":{"kind":"arxiv","id":"2304.00419","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2304.00419","created_at":"2026-07-05T05:57:02Z"},{"alias_kind":"arxiv_version","alias_value":"2304.00419v1","created_at":"2026-07-05T05:57:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2304.00419","created_at":"2026-07-05T05:57:02Z"},{"alias_kind":"pith_short_12","alias_value":"NDQO6ACKZ3G5","created_at":"2026-07-05T05:57:02Z"},{"alias_kind":"pith_short_16","alias_value":"NDQO6ACKZ3G5C32C","created_at":"2026-07-05T05:57:02Z"},{"alias_kind":"pith_short_8","alias_value":"NDQO6ACK","created_at":"2026-07-05T05:57:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:NDQO6ACKZ3G5C32CAYQPEHHY7Q","target":"record","payload":{"canonical_record":{"source":{"id":"2304.00419","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-04-02T00:58:29Z","cross_cats_sorted":["cs.AI","cs.DS"],"title_canon_sha256":"65e50a8e25bf4274c29971b92a2833b106350b33008604ac19364d106099b0c8","abstract_canon_sha256":"0f8f14a996f6ecab179ae341e32f6312e575fcbdcc595fe365742f537617663c"},"schema_version":"1.0"},"canonical_sha256":"68e0ef004acecdd16f420620f21cf8fc16339aff9ab7e8bf40f58676b80b49e8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:57:02.429637Z","signature_b64":"Rw2vDv1YRvKuAHwNzlgxSlHOVSDTfIPl6kSv2hHDUW2QWLrlz6FDcYL9tTVclKmCa7UMzXKcL8TOT6OgdUrGCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68e0ef004acecdd16f420620f21cf8fc16339aff9ab7e8bf40f58676b80b49e8","last_reissued_at":"2026-07-05T05:57:02.429249Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:57:02.429249Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2304.00419","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":"2026-07-05T05:57:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XL7u+0IudE8wETn7QSYt9hDdlzuWxb0McLVjovf1J1C30f+0gUPRcCeQKwXO78qgOwg3XAXrkNl2tNqtbMwqDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T13:45:43.890967Z"},"content_sha256":"de31dda974cefbeda8154af1901c741220144115d406dcc53090e303fe749335","schema_version":"1.0","event_id":"sha256:de31dda974cefbeda8154af1901c741220144115d406dcc53090e303fe749335"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:NDQO6ACKZ3G5C32CAYQPEHHY7Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mini-batch $k$-means terminates within $O(d/\\epsilon)$ iterations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","cs.DS"],"primary_cat":"cs.LG","authors_text":"Gregory Schwartzman","submitted_at":"2023-04-02T00:58:29Z","abstract_excerpt":"We answer the question: \"Does local progress (on batches) imply global progress (on the entire dataset) for mini-batch $k$-means?\". Specifically, we consider mini-batch $k$-means which terminates only when the improvement in the quality of the clustering on the sampled batch is below some threshold.\n  Although at first glance it appears that this algorithm might execute forever, we answer the above question in the affirmative and show that if the batch is of size $\\tilde{\\Omega}((d/\\epsilon)^2)$, it must terminate within $O(d/\\epsilon)$ iterations with high probability, where $d$ is the dimens"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.00419","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/2304.00419/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":"2026-07-05T05:57:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QB2hTkK0PLVHteiOkYUwcjXyi2HNLjFpwaMvvuE8mzWr//BBsK66VCK5iYDy97zSVGo+KIFkmRAiQd6JACZZCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T13:45:43.891360Z"},"content_sha256":"ae6d827bc1fe27a34e472119aac532175b939c0631840fbea727fc8bbad02dfb","schema_version":"1.0","event_id":"sha256:ae6d827bc1fe27a34e472119aac532175b939c0631840fbea727fc8bbad02dfb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NDQO6ACKZ3G5C32CAYQPEHHY7Q/bundle.json","state_url":"https://pith.science/pith/NDQO6ACKZ3G5C32CAYQPEHHY7Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NDQO6ACKZ3G5C32CAYQPEHHY7Q/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":"2026-07-07T13:45:43Z","links":{"resolver":"https://pith.science/pith/NDQO6ACKZ3G5C32CAYQPEHHY7Q","bundle":"https://pith.science/pith/NDQO6ACKZ3G5C32CAYQPEHHY7Q/bundle.json","state":"https://pith.science/pith/NDQO6ACKZ3G5C32CAYQPEHHY7Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NDQO6ACKZ3G5C32CAYQPEHHY7Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:NDQO6ACKZ3G5C32CAYQPEHHY7Q","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":"0f8f14a996f6ecab179ae341e32f6312e575fcbdcc595fe365742f537617663c","cross_cats_sorted":["cs.AI","cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-04-02T00:58:29Z","title_canon_sha256":"65e50a8e25bf4274c29971b92a2833b106350b33008604ac19364d106099b0c8"},"schema_version":"1.0","source":{"id":"2304.00419","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2304.00419","created_at":"2026-07-05T05:57:02Z"},{"alias_kind":"arxiv_version","alias_value":"2304.00419v1","created_at":"2026-07-05T05:57:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2304.00419","created_at":"2026-07-05T05:57:02Z"},{"alias_kind":"pith_short_12","alias_value":"NDQO6ACKZ3G5","created_at":"2026-07-05T05:57:02Z"},{"alias_kind":"pith_short_16","alias_value":"NDQO6ACKZ3G5C32C","created_at":"2026-07-05T05:57:02Z"},{"alias_kind":"pith_short_8","alias_value":"NDQO6ACK","created_at":"2026-07-05T05:57:02Z"}],"graph_snapshots":[{"event_id":"sha256:ae6d827bc1fe27a34e472119aac532175b939c0631840fbea727fc8bbad02dfb","target":"graph","created_at":"2026-07-05T05:57:02Z","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/2304.00419/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We answer the question: \"Does local progress (on batches) imply global progress (on the entire dataset) for mini-batch $k$-means?\". Specifically, we consider mini-batch $k$-means which terminates only when the improvement in the quality of the clustering on the sampled batch is below some threshold.\n  Although at first glance it appears that this algorithm might execute forever, we answer the above question in the affirmative and show that if the batch is of size $\\tilde{\\Omega}((d/\\epsilon)^2)$, it must terminate within $O(d/\\epsilon)$ iterations with high probability, where $d$ is the dimens","authors_text":"Gregory Schwartzman","cross_cats":["cs.AI","cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-04-02T00:58:29Z","title":"Mini-batch $k$-means terminates within $O(d/\\epsilon)$ iterations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.00419","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:de31dda974cefbeda8154af1901c741220144115d406dcc53090e303fe749335","target":"record","created_at":"2026-07-05T05:57:02Z","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":"0f8f14a996f6ecab179ae341e32f6312e575fcbdcc595fe365742f537617663c","cross_cats_sorted":["cs.AI","cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-04-02T00:58:29Z","title_canon_sha256":"65e50a8e25bf4274c29971b92a2833b106350b33008604ac19364d106099b0c8"},"schema_version":"1.0","source":{"id":"2304.00419","kind":"arxiv","version":1}},"canonical_sha256":"68e0ef004acecdd16f420620f21cf8fc16339aff9ab7e8bf40f58676b80b49e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68e0ef004acecdd16f420620f21cf8fc16339aff9ab7e8bf40f58676b80b49e8","first_computed_at":"2026-07-05T05:57:02.429249Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:57:02.429249Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Rw2vDv1YRvKuAHwNzlgxSlHOVSDTfIPl6kSv2hHDUW2QWLrlz6FDcYL9tTVclKmCa7UMzXKcL8TOT6OgdUrGCA==","signature_status":"signed_v1","signed_at":"2026-07-05T05:57:02.429637Z","signed_message":"canonical_sha256_bytes"},"source_id":"2304.00419","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de31dda974cefbeda8154af1901c741220144115d406dcc53090e303fe749335","sha256:ae6d827bc1fe27a34e472119aac532175b939c0631840fbea727fc8bbad02dfb"],"state_sha256":"a5a355bedb2b3b4bc70344830e9feac5cd0fd5d55cb79c907a9f4b3eb6a76ffa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+LdLTjngHgMULytR78Eoqfnl3+axfVX9NEOQxf+LBdq0wrmYetdUGMirQnh/671iv6bq19ERnugxHzaxV8TXBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T13:45:43.893356Z","bundle_sha256":"46670a0611d3eb6dc18157a9502e920cf126ac766bb729a6c849e21c323a9241"}}