{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:VRXHF5WFDDOXAU5GCHGCSDL5Z4","short_pith_number":"pith:VRXHF5WF","schema_version":"1.0","canonical_sha256":"ac6e72f6c518dd7053a611cc290d7dcf19c229ee40ed80ad61c53b43a4c44e41","source":{"kind":"arxiv","id":"2004.12511","version":2},"attestation_state":"computed","paper":{"title":"Hierarchical Low-Rank Approximation of Regularized Wasserstein Distance","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Mohammad Motamed","submitted_at":"2020-04-27T00:02:59Z","abstract_excerpt":"Sinkhorn divergence is a measure of dissimilarity between two probability measures. It is obtained through adding an entropic regularization term to Kantorovich's optimal transport problem and can hence be viewed as an entropically regularized Wasserstein distance. Given two discrete probability vectors in the $n$-simplex and supported on two bounded spaces in ${\\mathbb R}^d$, we present a fast method for computing Sinkhorn divergence when the cost matrix can be decomposed into a $d$-term sum of asymptotically smooth Kronecker product factors. The method combines Sinkhorn's matrix scaling iter"},"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":"2004.12511","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2020-04-27T00:02:59Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"c1bfb9fadbaf0b8f1a21efb9bbb2bc11098d3d826bae171b00481227cbf3b782","abstract_canon_sha256":"cff49e8c461d5ea84d09c08a80ae2c6174234a7bcb99d04ff1d2f9ca163e9a89"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:59:25.379916Z","signature_b64":"VzLNc0Z/qSJpT5WywFE47Svj9MZrKdILNO2rhRndfM115jwNmcf2yOIRKe9D6C3V/Zh9co0d+JPQkhi4di7iDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac6e72f6c518dd7053a611cc290d7dcf19c229ee40ed80ad61c53b43a4c44e41","last_reissued_at":"2026-07-05T00:59:25.379399Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:59:25.379399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hierarchical Low-Rank Approximation of Regularized Wasserstein Distance","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Mohammad Motamed","submitted_at":"2020-04-27T00:02:59Z","abstract_excerpt":"Sinkhorn divergence is a measure of dissimilarity between two probability measures. It is obtained through adding an entropic regularization term to Kantorovich's optimal transport problem and can hence be viewed as an entropically regularized Wasserstein distance. Given two discrete probability vectors in the $n$-simplex and supported on two bounded spaces in ${\\mathbb R}^d$, we present a fast method for computing Sinkhorn divergence when the cost matrix can be decomposed into a $d$-term sum of asymptotically smooth Kronecker product factors. The method combines Sinkhorn's matrix scaling iter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2004.12511","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2004.12511/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2004.12511","created_at":"2026-07-05T00:59:25.379461+00:00"},{"alias_kind":"arxiv_version","alias_value":"2004.12511v2","created_at":"2026-07-05T00:59:25.379461+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2004.12511","created_at":"2026-07-05T00:59:25.379461+00:00"},{"alias_kind":"pith_short_12","alias_value":"VRXHF5WFDDOX","created_at":"2026-07-05T00:59:25.379461+00:00"},{"alias_kind":"pith_short_16","alias_value":"VRXHF5WFDDOXAU5G","created_at":"2026-07-05T00:59:25.379461+00:00"},{"alias_kind":"pith_short_8","alias_value":"VRXHF5WF","created_at":"2026-07-05T00:59:25.379461+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VRXHF5WFDDOXAU5GCHGCSDL5Z4","json":"https://pith.science/pith/VRXHF5WFDDOXAU5GCHGCSDL5Z4.json","graph_json":"https://pith.science/api/pith-number/VRXHF5WFDDOXAU5GCHGCSDL5Z4/graph.json","events_json":"https://pith.science/api/pith-number/VRXHF5WFDDOXAU5GCHGCSDL5Z4/events.json","paper":"https://pith.science/paper/VRXHF5WF"},"agent_actions":{"view_html":"https://pith.science/pith/VRXHF5WFDDOXAU5GCHGCSDL5Z4","download_json":"https://pith.science/pith/VRXHF5WFDDOXAU5GCHGCSDL5Z4.json","view_paper":"https://pith.science/paper/VRXHF5WF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2004.12511&json=true","fetch_graph":"https://pith.science/api/pith-number/VRXHF5WFDDOXAU5GCHGCSDL5Z4/graph.json","fetch_events":"https://pith.science/api/pith-number/VRXHF5WFDDOXAU5GCHGCSDL5Z4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VRXHF5WFDDOXAU5GCHGCSDL5Z4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VRXHF5WFDDOXAU5GCHGCSDL5Z4/action/storage_attestation","attest_author":"https://pith.science/pith/VRXHF5WFDDOXAU5GCHGCSDL5Z4/action/author_attestation","sign_citation":"https://pith.science/pith/VRXHF5WFDDOXAU5GCHGCSDL5Z4/action/citation_signature","submit_replication":"https://pith.science/pith/VRXHF5WFDDOXAU5GCHGCSDL5Z4/action/replication_record"}},"created_at":"2026-07-05T00:59:25.379461+00:00","updated_at":"2026-07-05T00:59:25.379461+00:00"}