{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:64VZZKRAOJO7ACNUZFVNVPE5AF","short_pith_number":"pith:64VZZKRA","schema_version":"1.0","canonical_sha256":"f72b9caa20725df009b4c96adabc9d0175faed28cc35d31efc053f4bd93f4623","source":{"kind":"arxiv","id":"2606.27767","version":1},"attestation_state":"computed","paper":{"title":"Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"cs.LG","authors_text":"Cl\\'ement Bonet, Pierre-Cyril Aubin-Frankowski, Youssef Mroueh","submitted_at":"2026-06-26T06:53:03Z","abstract_excerpt":"Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumpti"},"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":"2606.27767","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-26T06:53:03Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"9fe92f4aac14233621f899f60241812583d92866f01d14778d422eccb0830cb0","abstract_canon_sha256":"93e05e7eab9f69bd4afdef9e8556f0b01dc5e596ba6c719b7971238043a213db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-29T01:14:47.812014Z","signature_b64":"23AbSQKYagWTdoJuSeuZl2x1sn3plqACwouJLd6aaMlU1lqVEMgFbrFmKx3XK8pwniSxuELHzxvbDfY/lqWHAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f72b9caa20725df009b4c96adabc9d0175faed28cc35d31efc053f4bd93f4623","last_reissued_at":"2026-06-29T01:14:47.811650Z","signature_status":"signed_v1","first_computed_at":"2026-06-29T01:14:47.811650Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"cs.LG","authors_text":"Cl\\'ement Bonet, Pierre-Cyril Aubin-Frankowski, Youssef Mroueh","submitted_at":"2026-06-26T06:53:03Z","abstract_excerpt":"Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumpti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27767","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.27767/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":"2606.27767","created_at":"2026-06-29T01:14:47.811705+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.27767v1","created_at":"2026-06-29T01:14:47.811705+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27767","created_at":"2026-06-29T01:14:47.811705+00:00"},{"alias_kind":"pith_short_12","alias_value":"64VZZKRAOJO7","created_at":"2026-06-29T01:14:47.811705+00:00"},{"alias_kind":"pith_short_16","alias_value":"64VZZKRAOJO7ACNU","created_at":"2026-06-29T01:14:47.811705+00:00"},{"alias_kind":"pith_short_8","alias_value":"64VZZKRA","created_at":"2026-06-29T01:14:47.811705+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/64VZZKRAOJO7ACNUZFVNVPE5AF","json":"https://pith.science/pith/64VZZKRAOJO7ACNUZFVNVPE5AF.json","graph_json":"https://pith.science/api/pith-number/64VZZKRAOJO7ACNUZFVNVPE5AF/graph.json","events_json":"https://pith.science/api/pith-number/64VZZKRAOJO7ACNUZFVNVPE5AF/events.json","paper":"https://pith.science/paper/64VZZKRA"},"agent_actions":{"view_html":"https://pith.science/pith/64VZZKRAOJO7ACNUZFVNVPE5AF","download_json":"https://pith.science/pith/64VZZKRAOJO7ACNUZFVNVPE5AF.json","view_paper":"https://pith.science/paper/64VZZKRA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.27767&json=true","fetch_graph":"https://pith.science/api/pith-number/64VZZKRAOJO7ACNUZFVNVPE5AF/graph.json","fetch_events":"https://pith.science/api/pith-number/64VZZKRAOJO7ACNUZFVNVPE5AF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/64VZZKRAOJO7ACNUZFVNVPE5AF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/64VZZKRAOJO7ACNUZFVNVPE5AF/action/storage_attestation","attest_author":"https://pith.science/pith/64VZZKRAOJO7ACNUZFVNVPE5AF/action/author_attestation","sign_citation":"https://pith.science/pith/64VZZKRAOJO7ACNUZFVNVPE5AF/action/citation_signature","submit_replication":"https://pith.science/pith/64VZZKRAOJO7ACNUZFVNVPE5AF/action/replication_record"}},"created_at":"2026-06-29T01:14:47.811705+00:00","updated_at":"2026-06-29T01:14:47.811705+00:00"}