{"paper":{"title":"Learning Unbiased Permutations via Flow Matching","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A flow matching model learns multimodal permutations by projecting trajectories onto the space of unit row and column sum matrices.","cross_cats":["cs.AI"],"primary_cat":"cs.LG","authors_text":"Carla P. Gomes, Yimeng Min","submitted_at":"2026-05-16T02:10:35Z","abstract_excerpt":"Learning permutations is fundamental to sorting, ranking, and matching, but existing differentiable methods based on entropy-regularized Sinkhorn produce a single softened solution and collapse under ambiguity. We present PermFlow, a conditional flow matching framework that operates directly on the affine subspace of matrices with unit row and column sums. A closed-form tangent-space projector preserves these constraints exactly along every trajectory, by construction rather than through iterative correction, and a nearest-target coupling routes distinct noisy initializations toward distinct v"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A conditional flow matching framework that operates directly on the affine subspace of matrices with unit row and column sums. A closed-form tangent-space projector preserves these constraints exactly along every trajectory by construction rather than through iterative correction, and a nearest-target coupling routes distinct noisy initializations toward distinct valid permutations, enabling the model to capture multimodal permutation distributions rather than collapsing them to a single mode.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the closed-form tangent-space projector can be computed efficiently and remains numerically stable for the chosen flow schedules, and that the nearest-target coupling strategy is sufficient to train the model to distinguish and recover distinct modes without requiring extra regularization or assumptions on the underlying data distribution.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"PermFlow applies conditional flow matching on the affine subspace of doubly stochastic matrices with a closed-form tangent projector and nearest-target coupling to capture multimodal permutation distributions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A flow matching model learns multimodal permutations by projecting trajectories onto the space of unit row and column sum matrices.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b44f748da307a8c0ba393bc13aa21985759112a24d0aa32e497654685aef2c17"},"source":{"id":"2605.16755","kind":"arxiv","version":1},"verdict":{"id":"f375d5c7-3741-4b64-8ae9-2702a497673f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:50:12.109022Z","strongest_claim":"A conditional flow matching framework that operates directly on the affine subspace of matrices with unit row and column sums. A closed-form tangent-space projector preserves these constraints exactly along every trajectory by construction rather than through iterative correction, and a nearest-target coupling routes distinct noisy initializations toward distinct valid permutations, enabling the model to capture multimodal permutation distributions rather than collapsing them to a single mode.","one_line_summary":"PermFlow applies conditional flow matching on the affine subspace of doubly stochastic matrices with a closed-form tangent projector and nearest-target coupling to capture multimodal permutation distributions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the closed-form tangent-space projector can be computed efficiently and remains numerically stable for the chosen flow schedules, and that the nearest-target coupling strategy is sufficient to train the model to distinguish and recover distinct modes without requiring extra regularization or assumptions on the underlying data distribution.","pith_extraction_headline":"A flow matching model learns multimodal permutations by projecting trajectories onto the space of unit row and column sum matrices."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16755/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:01:19.795486Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:01:02.915877Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T19:01:56.323429Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.454675Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b522754934c62d4653788999bd5f3a26c1943a6c15bc09a47fdaad732a3f0151"},"references":{"count":21,"sample":[{"doi":"","year":null,"title":"Improving and generalizing flow-based generative models with minibatch optimal transport","work_id":"7b689852-603d-46be-aaa5-c38325ba2182","ref_index":1,"cited_arxiv_id":"2302.00482","is_internal_anchor":true},{"doi":"","year":null,"title":"Forty-first international conference on machine learning , year=","work_id":"e8b2b5ad-cc79-4c1f-8154-a6c759a76754","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Advances in Neural Information Processing Systems , volume=","work_id":"f9f7887c-2cac-4e56-87eb-08a3321cfd80","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Advances in Neural Information Processing Systems , volume=","work_id":"91b427e6-acdf-4c9e-a946-983589a7f6e3","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Flow matching on general geometries","work_id":"ffbe7c63-9aed-405e-b074-a6412192fc38","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":21,"snapshot_sha256":"4edd6626ebf49ae355dc68cfece1c1e583cd3925e68e5b9a2f032fc64596710c","internal_anchors":5},"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"}