{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:NRBTFBE6SHM7747MHGQS3XIKR7","short_pith_number":"pith:NRBTFBE6","canonical_record":{"source":{"id":"1807.07924","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-07-20T16:32:22Z","cross_cats_sorted":["cs.CG","stat.ML"],"title_canon_sha256":"2dd3579096eb01a6a2526e4d07892c515fc8b621b1b2c69b849d7c91558ef254","abstract_canon_sha256":"a7325b073a75f7a9452eff19a3f248a06af8463075d398f62391eb021f0687cc"},"schema_version":"1.0"},"canonical_sha256":"6c4332849e91d9fff3ec39a12ddd0a8fcd36ec1bfea20cc95372687924dee709","source":{"kind":"arxiv","id":"1807.07924","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.07924","created_at":"2026-05-18T00:10:15Z"},{"alias_kind":"arxiv_version","alias_value":"1807.07924v1","created_at":"2026-05-18T00:10:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.07924","created_at":"2026-05-18T00:10:15Z"},{"alias_kind":"pith_short_12","alias_value":"NRBTFBE6SHM7","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NRBTFBE6SHM7747M","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NRBTFBE6","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:NRBTFBE6SHM7747MHGQS3XIKR7","target":"record","payload":{"canonical_record":{"source":{"id":"1807.07924","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-07-20T16:32:22Z","cross_cats_sorted":["cs.CG","stat.ML"],"title_canon_sha256":"2dd3579096eb01a6a2526e4d07892c515fc8b621b1b2c69b849d7c91558ef254","abstract_canon_sha256":"a7325b073a75f7a9452eff19a3f248a06af8463075d398f62391eb021f0687cc"},"schema_version":"1.0"},"canonical_sha256":"6c4332849e91d9fff3ec39a12ddd0a8fcd36ec1bfea20cc95372687924dee709","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:15.287122Z","signature_b64":"CdxxlvWJVMkhgY+NCNQG6OC5A6AujYwZBy5dI1z/4Y1n8ebI6yAJKOr0/RoElfnJkD3AnlXVIXrdT7gbojZiBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c4332849e91d9fff3ec39a12ddd0a8fcd36ec1bfea20cc95372687924dee709","last_reissued_at":"2026-05-18T00:10:15.286401Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:15.286401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.07924","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":"2026-05-18T00:10:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D0ZMTZokX9ImeV1SmSLM0wuoTWEn3yh5vDpuOs0uBrebZO1dhJvhwXuvNx3CD65orxckj3IewRxj/QyMduSpAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:57:39.485376Z"},"content_sha256":"801421e0fb53844caaa78455117a7400924c45b2d960d28fb3129e54ac421a65","schema_version":"1.0","event_id":"sha256:801421e0fb53844caaa78455117a7400924c45b2d960d28fb3129e54ac421a65"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:NRBTFBE6SHM7747MHGQS3XIKR7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Optimal Bounds on the VC-dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","stat.ML"],"primary_cat":"cs.LG","authors_text":"Andrey Kupavskii, Monika Csikos, Nabil H. Mustafa","submitted_at":"2018-07-20T16:32:22Z","abstract_excerpt":"The VC-dimension of a set system is a way to capture its complexity and has been a key parameter studied extensively in machine learning and geometry communities. In this paper, we resolve two longstanding open problems on bounding the VC-dimension of two fundamental set systems: $k$-fold unions/intersections of half-spaces, and the simplices set system. Among other implications, it settles an open question in machine learning that was first studied in the 1989 foundational paper of Blumer, Ehrenfeucht, Haussler and Warmuth as well as by Eisenstat and Angluin and Johnson."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.07924","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":"2026-05-18T00:10:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JmzsdmMGpqRtgpwh7gVJdf/z94xhxpMPohQSDH93q5+sIeVTX2F8PNuESJCEheuTPUVjZDcIT03T+R+7alUrAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:57:39.485963Z"},"content_sha256":"adeb3250158b694bafe436abf7b0f7b5d9604e566ab944e76d42e5a2f7d8203e","schema_version":"1.0","event_id":"sha256:adeb3250158b694bafe436abf7b0f7b5d9604e566ab944e76d42e5a2f7d8203e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NRBTFBE6SHM7747MHGQS3XIKR7/bundle.json","state_url":"https://pith.science/pith/NRBTFBE6SHM7747MHGQS3XIKR7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NRBTFBE6SHM7747MHGQS3XIKR7/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":"2026-05-31T09:57:39Z","links":{"resolver":"https://pith.science/pith/NRBTFBE6SHM7747MHGQS3XIKR7","bundle":"https://pith.science/pith/NRBTFBE6SHM7747MHGQS3XIKR7/bundle.json","state":"https://pith.science/pith/NRBTFBE6SHM7747MHGQS3XIKR7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NRBTFBE6SHM7747MHGQS3XIKR7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NRBTFBE6SHM7747MHGQS3XIKR7","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":"a7325b073a75f7a9452eff19a3f248a06af8463075d398f62391eb021f0687cc","cross_cats_sorted":["cs.CG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-07-20T16:32:22Z","title_canon_sha256":"2dd3579096eb01a6a2526e4d07892c515fc8b621b1b2c69b849d7c91558ef254"},"schema_version":"1.0","source":{"id":"1807.07924","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.07924","created_at":"2026-05-18T00:10:15Z"},{"alias_kind":"arxiv_version","alias_value":"1807.07924v1","created_at":"2026-05-18T00:10:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.07924","created_at":"2026-05-18T00:10:15Z"},{"alias_kind":"pith_short_12","alias_value":"NRBTFBE6SHM7","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NRBTFBE6SHM7747M","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NRBTFBE6","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:adeb3250158b694bafe436abf7b0f7b5d9604e566ab944e76d42e5a2f7d8203e","target":"graph","created_at":"2026-05-18T00:10:15Z","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 VC-dimension of a set system is a way to capture its complexity and has been a key parameter studied extensively in machine learning and geometry communities. In this paper, we resolve two longstanding open problems on bounding the VC-dimension of two fundamental set systems: $k$-fold unions/intersections of half-spaces, and the simplices set system. Among other implications, it settles an open question in machine learning that was first studied in the 1989 foundational paper of Blumer, Ehrenfeucht, Haussler and Warmuth as well as by Eisenstat and Angluin and Johnson.","authors_text":"Andrey Kupavskii, Monika Csikos, Nabil H. Mustafa","cross_cats":["cs.CG","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-07-20T16:32:22Z","title":"Optimal Bounds on the VC-dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.07924","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":"sha256:801421e0fb53844caaa78455117a7400924c45b2d960d28fb3129e54ac421a65","target":"record","created_at":"2026-05-18T00:10:15Z","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":"a7325b073a75f7a9452eff19a3f248a06af8463075d398f62391eb021f0687cc","cross_cats_sorted":["cs.CG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-07-20T16:32:22Z","title_canon_sha256":"2dd3579096eb01a6a2526e4d07892c515fc8b621b1b2c69b849d7c91558ef254"},"schema_version":"1.0","source":{"id":"1807.07924","kind":"arxiv","version":1}},"canonical_sha256":"6c4332849e91d9fff3ec39a12ddd0a8fcd36ec1bfea20cc95372687924dee709","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6c4332849e91d9fff3ec39a12ddd0a8fcd36ec1bfea20cc95372687924dee709","first_computed_at":"2026-05-18T00:10:15.286401Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:15.286401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CdxxlvWJVMkhgY+NCNQG6OC5A6AujYwZBy5dI1z/4Y1n8ebI6yAJKOr0/RoElfnJkD3AnlXVIXrdT7gbojZiBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:15.287122Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.07924","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:801421e0fb53844caaa78455117a7400924c45b2d960d28fb3129e54ac421a65","sha256:adeb3250158b694bafe436abf7b0f7b5d9604e566ab944e76d42e5a2f7d8203e"],"state_sha256":"0d4bb43595eaf14ffeafd0362cc6a4718e07eca5397ac18d31c7a2e257cfc39b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IMllUxk+s0H2QMyFlv5BTXLytYK21SrPDhs5Q5BL3UMvDWeYe6VbUfYsRfsbnYGxmzFRP6eFAI/CsYxBYZFjDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T09:57:39.488908Z","bundle_sha256":"b57c3bdff79d0a736d90674d825a8e1541d5ed251a51180e99e270da0c5bb3b9"}}