{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:Z2LZYMWHDG2MJLIQSRX5IO22TF","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":"203d0e277df35fcb193ab416396d273a4f143a22e697d9e626fc52bae9ae69b6","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-01T02:44:53Z","title_canon_sha256":"5bf51d82bbd29d23e4e90dd54b9bdfc464415409ccc76f1f398282dba8f35ea4"},"schema_version":"1.0","source":{"id":"1707.00087","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.00087","created_at":"2026-05-18T00:41:06Z"},{"alias_kind":"arxiv_version","alias_value":"1707.00087v1","created_at":"2026-05-18T00:41:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.00087","created_at":"2026-05-18T00:41:06Z"},{"alias_kind":"pith_short_12","alias_value":"Z2LZYMWHDG2M","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"Z2LZYMWHDG2MJLIQ","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"Z2LZYMWH","created_at":"2026-05-18T12:31:59Z"}],"graph_snapshots":[{"event_id":"sha256:d4a3b52d64bd0268cfd97cf68557c8d416fc1e52133fa6d560f5d5de4f46c518","target":"graph","created_at":"2026-05-18T00:41:06Z","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 Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the empirical measure obtained from $n$ independent samples from $\\mu$ approaches $\\mu$ in the Wasserstein distance of any order. We prove sharp asymptotic and finite-sample results for this rate of convergence for general measures on general compact metric spaces. Our finite-sample results show the existence of multi-scale behavior, where measures can exhibit r","authors_text":"Francis Bach, Jonathan Weed","cross_cats":["math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-01T02:44:53Z","title":"Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.00087","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:aea163e72130454e597343083c947c7b7783612ca1a3847d2a35c0f900251f46","target":"record","created_at":"2026-05-18T00:41:06Z","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":"203d0e277df35fcb193ab416396d273a4f143a22e697d9e626fc52bae9ae69b6","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-01T02:44:53Z","title_canon_sha256":"5bf51d82bbd29d23e4e90dd54b9bdfc464415409ccc76f1f398282dba8f35ea4"},"schema_version":"1.0","source":{"id":"1707.00087","kind":"arxiv","version":1}},"canonical_sha256":"ce979c32c719b4c4ad10946fd43b5a9951c8f88ce8cee1dd48e121cf2071daf6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce979c32c719b4c4ad10946fd43b5a9951c8f88ce8cee1dd48e121cf2071daf6","first_computed_at":"2026-05-18T00:41:06.529265Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:06.529265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pt27RicXrJTWKugggzoq3uu7KCTjL0iQ9341q74s7ETOynNC0O3Swv2XXGtxoKvwSfQZwpTYIezNgSNl6z7uDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:06.529929Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.00087","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aea163e72130454e597343083c947c7b7783612ca1a3847d2a35c0f900251f46","sha256:d4a3b52d64bd0268cfd97cf68557c8d416fc1e52133fa6d560f5d5de4f46c518"],"state_sha256":"b6fa5a7d651426bbd7f32c64c1f8106950eeee2c61aa6a05add5a8991f335dae"}