{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:NCUCRR6G5IPWUFRTKUX33JHRGE","short_pith_number":"pith:NCUCRR6G","schema_version":"1.0","canonical_sha256":"68a828c7c6ea1f6a1633552fbda4f1313b5d21cff8068d8481b01af8a0d6212b","source":{"kind":"arxiv","id":"2411.09848","version":2},"attestation_state":"computed","paper":{"title":"Wasserstein Gradient Flows of MMD Functionals with Distance Kernels under Sobolev Regularization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gabriele Steidl, Nicolaj Rux, Richard Duong, Viktor Stein","submitted_at":"2024-11-14T23:48:39Z","abstract_excerpt":"We consider Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\\text{MMD}_K^2(\\cdot, \\nu)$ for positive and negative distance kernels $K(x,y) := \\pm |x-y|$ and given target measures $\\nu$ on $\\mathbb{R}$. Since in one dimension the Wasserstein space can be isometrically embedded into the cone $\\mathcal C(0,1) \\subset L_2(0,1)$ of quantile functions, Wasserstein gradient flows can be characterized by the solution of an associated Cauchy problem on $L_2(0,1)$. While for the negative kernel, the MMD functional is geodesically convex, this is not the case for the positive ke"},"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":"2411.09848","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-14T23:48:39Z","cross_cats_sorted":[],"title_canon_sha256":"6a56ea85fd771498b28acf5d59d927bc6e084ec3c277b66b4b23e205f8e1fbeb","abstract_canon_sha256":"8ac957ca91b4190af9624b33042f7d0d2e5cf42d81da703a46ea3ae8d859db0a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:58:20.408811Z","signature_b64":"1HyFYOd3UhCq/URzEnQgJ56E+R1bC1hZUMn/3Mw+z5iyVyIMUKQVAavpGMk5qFqJA3X+gQWLlk5orlEVFjd3Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68a828c7c6ea1f6a1633552fbda4f1313b5d21cff8068d8481b01af8a0d6212b","last_reissued_at":"2026-07-05T10:58:20.408270Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:58:20.408270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Wasserstein Gradient Flows of MMD Functionals with Distance Kernels under Sobolev Regularization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gabriele Steidl, Nicolaj Rux, Richard Duong, Viktor Stein","submitted_at":"2024-11-14T23:48:39Z","abstract_excerpt":"We consider Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\\text{MMD}_K^2(\\cdot, \\nu)$ for positive and negative distance kernels $K(x,y) := \\pm |x-y|$ and given target measures $\\nu$ on $\\mathbb{R}$. Since in one dimension the Wasserstein space can be isometrically embedded into the cone $\\mathcal C(0,1) \\subset L_2(0,1)$ of quantile functions, Wasserstein gradient flows can be characterized by the solution of an associated Cauchy problem on $L_2(0,1)$. While for the negative kernel, the MMD functional is geodesically convex, this is not the case for the positive ke"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.09848","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/2411.09848/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":"2411.09848","created_at":"2026-07-05T10:58:20.408338+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.09848v2","created_at":"2026-07-05T10:58:20.408338+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.09848","created_at":"2026-07-05T10:58:20.408338+00:00"},{"alias_kind":"pith_short_12","alias_value":"NCUCRR6G5IPW","created_at":"2026-07-05T10:58:20.408338+00:00"},{"alias_kind":"pith_short_16","alias_value":"NCUCRR6G5IPWUFRT","created_at":"2026-07-05T10:58:20.408338+00:00"},{"alias_kind":"pith_short_8","alias_value":"NCUCRR6G","created_at":"2026-07-05T10:58:20.408338+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/NCUCRR6G5IPWUFRTKUX33JHRGE","json":"https://pith.science/pith/NCUCRR6G5IPWUFRTKUX33JHRGE.json","graph_json":"https://pith.science/api/pith-number/NCUCRR6G5IPWUFRTKUX33JHRGE/graph.json","events_json":"https://pith.science/api/pith-number/NCUCRR6G5IPWUFRTKUX33JHRGE/events.json","paper":"https://pith.science/paper/NCUCRR6G"},"agent_actions":{"view_html":"https://pith.science/pith/NCUCRR6G5IPWUFRTKUX33JHRGE","download_json":"https://pith.science/pith/NCUCRR6G5IPWUFRTKUX33JHRGE.json","view_paper":"https://pith.science/paper/NCUCRR6G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.09848&json=true","fetch_graph":"https://pith.science/api/pith-number/NCUCRR6G5IPWUFRTKUX33JHRGE/graph.json","fetch_events":"https://pith.science/api/pith-number/NCUCRR6G5IPWUFRTKUX33JHRGE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NCUCRR6G5IPWUFRTKUX33JHRGE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NCUCRR6G5IPWUFRTKUX33JHRGE/action/storage_attestation","attest_author":"https://pith.science/pith/NCUCRR6G5IPWUFRTKUX33JHRGE/action/author_attestation","sign_citation":"https://pith.science/pith/NCUCRR6G5IPWUFRTKUX33JHRGE/action/citation_signature","submit_replication":"https://pith.science/pith/NCUCRR6G5IPWUFRTKUX33JHRGE/action/replication_record"}},"created_at":"2026-07-05T10:58:20.408338+00:00","updated_at":"2026-07-05T10:58:20.408338+00:00"}