{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:2FIQJTGYNUFHGD4AFRSIIGM5AV","short_pith_number":"pith:2FIQJTGY","canonical_record":{"source":{"id":"1209.1121","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-09-05T21:18:03Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"ffe25e77cc2af7dedb896810e8c3d6672ee3615b740e9c37ffe24c74ad172450","abstract_canon_sha256":"d8a393146188a4e8329bec517cdf33cd58d4ad7e192f5746d0b3b68c6b812227"},"schema_version":"1.0"},"canonical_sha256":"d15104ccd86d0a730f802c6484199d057c9c93d37a1e660488d602dacb1a1436","source":{"kind":"arxiv","id":"1209.1121","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.1121","created_at":"2026-05-18T02:20:50Z"},{"alias_kind":"arxiv_version","alias_value":"1209.1121v4","created_at":"2026-05-18T02:20:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.1121","created_at":"2026-05-18T02:20:50Z"},{"alias_kind":"pith_short_12","alias_value":"2FIQJTGYNUFH","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2FIQJTGYNUFHGD4A","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2FIQJTGY","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:2FIQJTGYNUFHGD4AFRSIIGM5AV","target":"record","payload":{"canonical_record":{"source":{"id":"1209.1121","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-09-05T21:18:03Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"ffe25e77cc2af7dedb896810e8c3d6672ee3615b740e9c37ffe24c74ad172450","abstract_canon_sha256":"d8a393146188a4e8329bec517cdf33cd58d4ad7e192f5746d0b3b68c6b812227"},"schema_version":"1.0"},"canonical_sha256":"d15104ccd86d0a730f802c6484199d057c9c93d37a1e660488d602dacb1a1436","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:50.462678Z","signature_b64":"+uIXU9Wocl5Nrc88wK+unlLYQ4c+CJKjk67j5yjQuxQW+4UAxu9Bbx7A7vF8kptL58q2DX04b05ITCWHJTW/Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d15104ccd86d0a730f802c6484199d057c9c93d37a1e660488d602dacb1a1436","last_reissued_at":"2026-05-18T02:20:50.462057Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:50.462057Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.1121","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:20:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"32Ia8hfSSR/VZiGwy7OVqailslzq6uqvsczeHRGUrigKUIq3s6+/hYy8kKhrwWyDRMR9FSQmbDaROeiHbq+kDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T05:00:17.352682Z"},"content_sha256":"a202fe5aa61756695d752dc93cf7d8bb0f8611306ef95b33e022ce4d37f96d58","schema_version":"1.0","event_id":"sha256:a202fe5aa61756695d752dc93cf7d8bb0f8611306ef95b33e022ce4d37f96d58"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:2FIQJTGYNUFHGD4AFRSIIGM5AV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Learning Manifolds with K-Means and K-Flats","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Guillermo D. Canas, Lorenzo Rosasco, Tomaso Poggio","submitted_at":"2012-09-05T21:18:03Z","abstract_excerpt":"We study the problem of estimating a manifold from random samples. In particular, we consider piecewise constant and piecewise linear estimators induced by k-means and k-flats, and analyze their performance. We extend previous results for k-means in two separate directions. First, we provide new results for k-means reconstruction on manifolds and, secondly, we prove reconstruction bounds for higher-order approximation (k-flats), for which no known results were previously available. While the results for k-means are novel, some of the technical tools are well-established in the literature. In t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1121","kind":"arxiv","version":4},"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":"2026-05-18T02:20:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UO9bYqRVxTBJp0Mwp5djGr0z3qHnhQUoRp9CPa+5nZrUQVYXqmogFWhrZJgM96R4w/oKpjIi95oBNTVgkM8HBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T05:00:17.353029Z"},"content_sha256":"12066389f25e912bd25ed414cc75fb87eb723c3fbe375a38ac48e015bbd2236e","schema_version":"1.0","event_id":"sha256:12066389f25e912bd25ed414cc75fb87eb723c3fbe375a38ac48e015bbd2236e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2FIQJTGYNUFHGD4AFRSIIGM5AV/bundle.json","state_url":"https://pith.science/pith/2FIQJTGYNUFHGD4AFRSIIGM5AV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2FIQJTGYNUFHGD4AFRSIIGM5AV/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":"2026-06-20T05:00:17Z","links":{"resolver":"https://pith.science/pith/2FIQJTGYNUFHGD4AFRSIIGM5AV","bundle":"https://pith.science/pith/2FIQJTGYNUFHGD4AFRSIIGM5AV/bundle.json","state":"https://pith.science/pith/2FIQJTGYNUFHGD4AFRSIIGM5AV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2FIQJTGYNUFHGD4AFRSIIGM5AV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:2FIQJTGYNUFHGD4AFRSIIGM5AV","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":"d8a393146188a4e8329bec517cdf33cd58d4ad7e192f5746d0b3b68c6b812227","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-09-05T21:18:03Z","title_canon_sha256":"ffe25e77cc2af7dedb896810e8c3d6672ee3615b740e9c37ffe24c74ad172450"},"schema_version":"1.0","source":{"id":"1209.1121","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.1121","created_at":"2026-05-18T02:20:50Z"},{"alias_kind":"arxiv_version","alias_value":"1209.1121v4","created_at":"2026-05-18T02:20:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.1121","created_at":"2026-05-18T02:20:50Z"},{"alias_kind":"pith_short_12","alias_value":"2FIQJTGYNUFH","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2FIQJTGYNUFHGD4A","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2FIQJTGY","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:12066389f25e912bd25ed414cc75fb87eb723c3fbe375a38ac48e015bbd2236e","target":"graph","created_at":"2026-05-18T02:20:50Z","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":"We study the problem of estimating a manifold from random samples. In particular, we consider piecewise constant and piecewise linear estimators induced by k-means and k-flats, and analyze their performance. We extend previous results for k-means in two separate directions. First, we provide new results for k-means reconstruction on manifolds and, secondly, we prove reconstruction bounds for higher-order approximation (k-flats), for which no known results were previously available. While the results for k-means are novel, some of the technical tools are well-established in the literature. In t","authors_text":"Guillermo D. Canas, Lorenzo Rosasco, Tomaso Poggio","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-09-05T21:18:03Z","title":"Learning Manifolds with K-Means and K-Flats"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1121","kind":"arxiv","version":4},"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:a202fe5aa61756695d752dc93cf7d8bb0f8611306ef95b33e022ce4d37f96d58","target":"record","created_at":"2026-05-18T02:20:50Z","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":"d8a393146188a4e8329bec517cdf33cd58d4ad7e192f5746d0b3b68c6b812227","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-09-05T21:18:03Z","title_canon_sha256":"ffe25e77cc2af7dedb896810e8c3d6672ee3615b740e9c37ffe24c74ad172450"},"schema_version":"1.0","source":{"id":"1209.1121","kind":"arxiv","version":4}},"canonical_sha256":"d15104ccd86d0a730f802c6484199d057c9c93d37a1e660488d602dacb1a1436","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d15104ccd86d0a730f802c6484199d057c9c93d37a1e660488d602dacb1a1436","first_computed_at":"2026-05-18T02:20:50.462057Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:20:50.462057Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+uIXU9Wocl5Nrc88wK+unlLYQ4c+CJKjk67j5yjQuxQW+4UAxu9Bbx7A7vF8kptL58q2DX04b05ITCWHJTW/Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:20:50.462678Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.1121","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a202fe5aa61756695d752dc93cf7d8bb0f8611306ef95b33e022ce4d37f96d58","sha256:12066389f25e912bd25ed414cc75fb87eb723c3fbe375a38ac48e015bbd2236e"],"state_sha256":"bde1381495819a496012d5f313c78c3c060ad810ec2e2f963eb1659dde17136d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1p8z/cgq9Uf4ZzvxLzqApImKGvvmcbHkL9IT7Copj8rGvBUbdpEfaURRXLWAiWEeFbrHstWBzn/S2dfFkAVdDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T05:00:17.354966Z","bundle_sha256":"772d7007d166d79493bfbfd8057d7f98cebf4712349ece3968e11dee57ba30bc"}}