{"paper":{"title":"Spherical Flows for Sampling Categorical Data","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"Spherical flows using the von Mises-Fisher distribution reduce categorical sequence sampling to solving a scalar ODE in cosine similarity.","cross_cats":["cs.CL","cs.LG"],"primary_cat":"stat.ML","authors_text":"Gabriele Steidl, Gregor Kornhardt, Jannis Chemseddine","submitted_at":"2026-05-07T03:34:00Z","abstract_excerpt":"We study the problem of learning generative models for discrete sequences in a continuous embedding space. Whereas prior approaches typically operate in Euclidean space or on the probability simplex, we instead work on the sphere $\\mathbb S^{d-1}$. There the von Mises-Fisher (vMF) distribution induces a natural noise process and admits a closed-form conditional score. The conditional velocity is in general intractable. Exploiting the radial symmetry of the vMF density we reduce the continuity equation on $\\mathbb S^{d-1}$ to a scalar ODE in the cosine similarity, whose unique bounded solution "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Exploiting the radial symmetry of the vMF density we reduce the continuity equation on S^{d-1} to a scalar ODE in the cosine similarity, whose unique bounded solution determines the velocity. The marginal velocity and marginal score on (S^{d-1})^L both decompose into posterior-weighted tangent sums.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the learned posterior (trained only by cross-entropy) is sufficiently accurate to produce stable posterior-weighted sums for both velocity and score during sampling on real discrete data.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Spherical vMF flows reduce the continuity equation on the sphere to a scalar ODE in cosine similarity, enabling posterior-weighted sampling of categorical sequences via cross-entropy trained posteriors.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Spherical flows using the von Mises-Fisher distribution reduce categorical sequence sampling to solving a scalar ODE in cosine similarity.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5dba8ad76b754c2ccce72edc53616688e31ad48c202b926eead51abc737cd001"},"source":{"id":"2605.05629","kind":"arxiv","version":3},"verdict":{"id":"46e8b717-95a6-4529-9d2a-318bbb9dc7be","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T01:33:40.029370Z","strongest_claim":"Exploiting the radial symmetry of the vMF density we reduce the continuity equation on S^{d-1} to a scalar ODE in the cosine similarity, whose unique bounded solution determines the velocity. The marginal velocity and marginal score on (S^{d-1})^L both decompose into posterior-weighted tangent sums.","one_line_summary":"Spherical vMF flows reduce the continuity equation on the sphere to a scalar ODE in cosine similarity, enabling posterior-weighted sampling of categorical sequences via cross-entropy trained posteriors.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the learned posterior (trained only by cross-entropy) is sufficiently accurate to produce stable posterior-weighted sums for both velocity and score during sampling on real discrete data.","pith_extraction_headline":"Spherical flows using the von Mises-Fisher distribution reduce categorical sequence sampling to solving a scalar ODE in cosine similarity."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.05629/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T14:02:04.714146Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-20T09:38:52.990973Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T20:01:19.897579Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T13:24:31.981767Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"acfaac3f4c82d2e922172bb038e481a681472287bb125c8285606cd14f158427"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"c78bc3072ef5ee2433ae6dc26c0c34cdaf3450b1d512b9ffb2061b55ed49c911"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}