{"paper":{"title":"Continuum-marginal optimal transport: a mesh-free kernel method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A reproducing kernel Hilbert space embedding of the weak continuity equation yields a sample-only objective for recovering minimum-energy velocity fields that match a continuum of probability marginals.","cross_cats":["cs.NA","math.NA","stat.ML"],"primary_cat":"math.OC","authors_text":"Yumiharu Nakano","submitted_at":"2026-04-27T09:33:52Z","abstract_excerpt":"In this paper we study continuum-marginal optimal transport. Given a time-continuous family of probability marginals, the problem is to recover the minimum-energy velocity field whose flow reproduces every marginal. This problem is the continuum limit of the classical two-marginal Benamou--Brenier formulation, and also the deterministic limit of the Nelson problem of stochastic optimal transport. We propose a practical mesh-free solver for this problem. The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial disc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial discretization. The velocity is parametrized by any linear-in-parameters dictionary or neural network, and is optimized by mini-batch stochastic methods. Synthetic experiments confirm that the method achieves accurate drift recovery and marginal consistency.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That embedding the weak continuity equation in an RKHS produces an objective whose minimizer recovers the true velocity field from finite samples without additional regularization or post-hoc corrections.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A reproducing kernel Hilbert space embedding yields a sample-only objective for solving continuum-marginal optimal transport without spatial discretization.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A reproducing kernel Hilbert space embedding of the weak continuity equation yields a sample-only objective for recovering minimum-energy velocity fields that match a continuum of probability marginals.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7605404ee5a00bcbc589a9bb9cc8cbf3ef0bf2c13c1ca33b1c96054c1d6f54f0"},"source":{"id":"2604.24226","kind":"arxiv","version":2},"verdict":{"id":"1d4933b6-d1b0-46ad-80e4-c122f7f3a719","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T02:39:28.080096Z","strongest_claim":"The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial discretization. The velocity is parametrized by any linear-in-parameters dictionary or neural network, and is optimized by mini-batch stochastic methods. Synthetic experiments confirm that the method achieves accurate drift recovery and marginal consistency.","one_line_summary":"A reproducing kernel Hilbert space embedding yields a sample-only objective for solving continuum-marginal optimal transport without spatial discretization.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That embedding the weak continuity equation in an RKHS produces an objective whose minimizer recovers the true velocity field from finite samples without additional regularization or post-hoc corrections.","pith_extraction_headline":"A reproducing kernel Hilbert space embedding of the weak continuity equation yields a sample-only objective for recovering minimum-energy velocity fields that match a continuum of probability marginals."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.24226/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T07:35:45.909032Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:17:08.829833Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"32d1345abf9ed08bc81f7c3567c1b8499c073f8e8b8a8a5949219013d832596c"},"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"}