{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:OV5JPABUM4YZJY7KDN4HJF457U","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":"4f98dc1c42b43c30efbb8eba72aac537b84223b7750ff33d5b388d106f04db3b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-02-25T08:35:52Z","title_canon_sha256":"b3e0ea25124de72c6920a42c6892445b6a0da616c68c5fa9cc8cc330c8c89499"},"schema_version":"1.0","source":{"id":"1902.09138","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.09138","created_at":"2026-05-17T23:52:46Z"},{"alias_kind":"arxiv_version","alias_value":"1902.09138v1","created_at":"2026-05-17T23:52:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.09138","created_at":"2026-05-17T23:52:46Z"},{"alias_kind":"pith_short_12","alias_value":"OV5JPABUM4YZ","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"OV5JPABUM4YZJY7K","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"OV5JPABU","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:873c4c432e6f388c368ec585558837b7869d8302e275ad46450eb3a749b349df","target":"graph","created_at":"2026-05-17T23:52:46Z","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":"Let $X$ be a closed subspace of a metric space $M$. Under mild hypotheses, one can estimate the Betti numbers of $X$ from a finite set $P \\subset M$ of points approximating $X$. In this paper, we show that one can also use $P$ to estimate much more detailed topological properties of $X$. These properties are computed via $A_\\infty$-structures, and are therefore related to the cup and Massey products of $X$, its loop space $\\Omega X$, its formality, linking numbers, etc.\n  Additionally, we study the following setting: given a continuous function $f \\colon Y \\longrightarrow \\mathbb R$ on a topol","authors_text":"Anastasios Stefanou, Francisco Belch\\'i","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-02-25T08:35:52Z","title":"$A_\\infty$ persistent homology estimates the topology from pointcloud datasets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.09138","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":"sha256:4e64167a0fd84273975ddbfbf1ecfe512faeb7ac0a14f0902291faa176bc5587","target":"record","created_at":"2026-05-17T23:52:46Z","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":"4f98dc1c42b43c30efbb8eba72aac537b84223b7750ff33d5b388d106f04db3b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-02-25T08:35:52Z","title_canon_sha256":"b3e0ea25124de72c6920a42c6892445b6a0da616c68c5fa9cc8cc330c8c89499"},"schema_version":"1.0","source":{"id":"1902.09138","kind":"arxiv","version":1}},"canonical_sha256":"757a978034673194e3ea1b7874979dfd365ef4ff0b603a6bc9e75cd8e9eee811","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"757a978034673194e3ea1b7874979dfd365ef4ff0b603a6bc9e75cd8e9eee811","first_computed_at":"2026-05-17T23:52:46.537351Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:52:46.537351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3YymDAPtcUBnpxyvyLakM85t9L82LXZUG258d0ZcpqyIfyyBlwFpM4tB6+cw/FdxZDrTgQVkmZpfcqVAgEU3Cw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:52:46.537859Z","signed_message":"canonical_sha256_bytes"},"source_id":"1902.09138","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4e64167a0fd84273975ddbfbf1ecfe512faeb7ac0a14f0902291faa176bc5587","sha256:873c4c432e6f388c368ec585558837b7869d8302e275ad46450eb3a749b349df"],"state_sha256":"986e0399af5cc18a9d2bbe4b698f8ab4705d8482670b665c9140e5b6af6f9f0e"}