{"paper":{"title":"Normalizing Flows on Quotient Manifolds via Boundary Quotients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Normalizing flows on the 3-sphere can be pushed forward via the covering map to approximate distributions on lens spaces.","cross_cats":["math.PR"],"primary_cat":"cs.LG","authors_text":"Benjamin Cai, William Ghanem","submitted_at":"2025-11-28T05:12:27Z","abstract_excerpt":"We introduce boundary quotients and present a framework for learning densities on manifolds that arise as boundary quotients of simpler domains. We show that this framework can be used to construct normalizing flows on quotient manifolds $N/G$, where a discrete group $G$ acts on $N$. We instantiate this construction for genus-$g$ surfaces $\\Sigma_g$. When $G$ is finite, we show applicability to symmetry aware learning; we demonstrate this on cyclic quotients of the 3-sphere. Experiments on lens spaces show that simple pre-quotient RealNVP models can achieve strong results while being substanti"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct pushforward distributions via the universal covering map rho: S^3 -> L(p;q) with the goal of approximating these distributions using flows on L(p;q). We highlight that our method deletes redundancies in the case of a symmetric S^3 distribution.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That flows trained on the covering space S^3 can be pushed forward to produce accurate approximations on L(p;q) without significant distortion or loss of expressivity due to the quotient map.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Approximates pushforwards of distributions on lens spaces by using normalizing flows on the covering 3-sphere, deleting redundancies for symmetric cases, and demonstrates on von Mises-Fisher densities plus a Z_12-symmetric benzene model.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Normalizing flows on the 3-sphere can be pushed forward via the covering map to approximate distributions on lens spaces.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3b3dd2222605e317e8fd3265dd07321da5d93b286a4867ca17e220132176985e"},"source":{"id":"2511.22882","kind":"arxiv","version":3},"verdict":{"id":"9be80654-9afd-47c6-8e0e-70ea3db7b288","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-17T04:18:43.035337Z","strongest_claim":"We construct pushforward distributions via the universal covering map rho: S^3 -> L(p;q) with the goal of approximating these distributions using flows on L(p;q). We highlight that our method deletes redundancies in the case of a symmetric S^3 distribution.","one_line_summary":"Approximates pushforwards of distributions on lens spaces by using normalizing flows on the covering 3-sphere, deleting redundancies for symmetric cases, and demonstrates on von Mises-Fisher densities plus a Z_12-symmetric benzene model.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That flows trained on the covering space S^3 can be pushed forward to produce accurate approximations on L(p;q) without significant distortion or loss of expressivity due to the quotient map.","pith_extraction_headline":"Normalizing flows on the 3-sphere can be pushed forward via the covering map to approximate distributions on lens spaces."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2511.22882/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":2,"snapshot_sha256":"a415bc67694fb77b98602832810270d18ec93dba1cca102b07f9a7d8905d7595"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}