{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:BK37LEYOCQZBXG2FQLE2HEHAW4","short_pith_number":"pith:BK37LEYO","schema_version":"1.0","canonical_sha256":"0ab7f5930e14321b9b4582c9a390e0b717e3a8df6ddf25870895d8e2a33cecc8","source":{"kind":"arxiv","id":"1209.1077","version":1},"attestation_state":"computed","paper":{"title":"Learning Probability Measures with respect to Optimal Transport Metrics","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","submitted_at":"2012-09-05T19:10:09Z","abstract_excerpt":"We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal quantization, and learning theory, we derive new probabilistic bounds for the performance of a classic algorithm in unsupervised learning (k-means), when used to produce a probability measure derived from the data. In the course of the analysis, we arrive at new lower bounds, as well as probabilistic upper bounds on the convergence rate of the empirical law o"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1209.1077","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-09-05T19:10:09Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"9c118edb0877d69ee77502a76082d01d5d50fd2436da99bdffda6038e36d98d4","abstract_canon_sha256":"9b5ca29c34761bd238478f9bfed07d66f993b67a921ac8244e7b634643312663"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:15.537947Z","signature_b64":"varCJYbEzUYaTI9npFe4S8Mg3pZzymfSwj+BN0QYWnFID1DxD9/d6VuMNf1qvpVU88+1jocZco+vruiJfWiODA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ab7f5930e14321b9b4582c9a390e0b717e3a8df6ddf25870895d8e2a33cecc8","last_reissued_at":"2026-05-18T03:46:15.536874Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:15.536874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Learning Probability Measures with respect to Optimal Transport Metrics","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","submitted_at":"2012-09-05T19:10:09Z","abstract_excerpt":"We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal quantization, and learning theory, we derive new probabilistic bounds for the performance of a classic algorithm in unsupervised learning (k-means), when used to produce a probability measure derived from the data. In the course of the analysis, we arrive at new lower bounds, as well as probabilistic upper bounds on the convergence rate of the empirical law o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1077","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1209.1077","created_at":"2026-05-18T03:46:15.536975+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.1077v1","created_at":"2026-05-18T03:46:15.536975+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.1077","created_at":"2026-05-18T03:46:15.536975+00:00"},{"alias_kind":"pith_short_12","alias_value":"BK37LEYOCQZB","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"BK37LEYOCQZBXG2F","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"BK37LEYO","created_at":"2026-05-18T12:27:01.376967+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.03240","citing_title":"On Model-Based Clustering With Entropic Optimal Transport","ref_index":269,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BK37LEYOCQZBXG2FQLE2HEHAW4","json":"https://pith.science/pith/BK37LEYOCQZBXG2FQLE2HEHAW4.json","graph_json":"https://pith.science/api/pith-number/BK37LEYOCQZBXG2FQLE2HEHAW4/graph.json","events_json":"https://pith.science/api/pith-number/BK37LEYOCQZBXG2FQLE2HEHAW4/events.json","paper":"https://pith.science/paper/BK37LEYO"},"agent_actions":{"view_html":"https://pith.science/pith/BK37LEYOCQZBXG2FQLE2HEHAW4","download_json":"https://pith.science/pith/BK37LEYOCQZBXG2FQLE2HEHAW4.json","view_paper":"https://pith.science/paper/BK37LEYO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.1077&json=true","fetch_graph":"https://pith.science/api/pith-number/BK37LEYOCQZBXG2FQLE2HEHAW4/graph.json","fetch_events":"https://pith.science/api/pith-number/BK37LEYOCQZBXG2FQLE2HEHAW4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BK37LEYOCQZBXG2FQLE2HEHAW4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BK37LEYOCQZBXG2FQLE2HEHAW4/action/storage_attestation","attest_author":"https://pith.science/pith/BK37LEYOCQZBXG2FQLE2HEHAW4/action/author_attestation","sign_citation":"https://pith.science/pith/BK37LEYOCQZBXG2FQLE2HEHAW4/action/citation_signature","submit_replication":"https://pith.science/pith/BK37LEYOCQZBXG2FQLE2HEHAW4/action/replication_record"}},"created_at":"2026-05-18T03:46:15.536975+00:00","updated_at":"2026-05-18T03:46:15.536975+00:00"}