{"paper":{"title":"Radial Compensation: Fixing Radius Distortion in Chart-Based Generative Models on Riemannian Manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Within isotropic scalar-Jacobian azimuthal charts no base distribution preserves geodesic-radial likelihoods, chart-invariant radial Fisher information, and tangent-space isotropy unless it takes the specific Radial Compensation form.","cross_cats":["cs.AI","cs.IT","math.DG","math.IT","stat.ML"],"primary_cat":"cs.LG","authors_text":"Marios Papamichalis, Regina Ruane","submitted_at":"2025-11-18T02:15:25Z","abstract_excerpt":"We study the base distribution in chart-based generative models on Riemannian manifolds. Standard methods sample in Euclidean tangent space and then map the sample to the manifold with a chart. This is convenient, but it changes the meaning of distance: the same tangent-space scale can correspond to different geodesic radii, i.e. shortest-path distances from a reference point on the manifold, under different charts, curvatures, and dimensions. Within isotropic, scalar-Jacobian azimuthal charts, we show that no base distribution can simultaneously preserve geodesic-radial likelihoods, chart-inv"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Within isotropic, scalar-Jacobian azimuthal charts, no base distribution can simultaneously preserve geodesic-radial likelihoods, chart-invariant radial Fisher information, and tangent-space isotropy unless it has a specific form, which we call Radial Compensation (RC).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis holds only inside the class of isotropic, scalar-Jacobian azimuthal charts; outside this class the impossibility result and the necessity of the RC form may not apply.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Radial Compensation derives a specific tangent-space base distribution that preserves geodesic-radial likelihoods and chart-invariant Fisher information while maintaining isotropy, decoupling statistical modeling from numerical chart choice in manifold generative models.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Within isotropic scalar-Jacobian azimuthal charts no base distribution preserves geodesic-radial likelihoods, chart-invariant radial Fisher information, and tangent-space isotropy unless it takes the specific Radial Compensation form.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4db59cffe2c1e1749a9fceff6f11ae13ea705cc690ac71a93b91ccf89396527b"},"source":{"id":"2511.14056","kind":"arxiv","version":2},"verdict":{"id":"0043c97f-7249-4d5e-917f-a93c57b20a24","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-17T20:19:29.362353Z","strongest_claim":"Within isotropic, scalar-Jacobian azimuthal charts, no base distribution can simultaneously preserve geodesic-radial likelihoods, chart-invariant radial Fisher information, and tangent-space isotropy unless it has a specific form, which we call Radial Compensation (RC).","one_line_summary":"Radial Compensation derives a specific tangent-space base distribution that preserves geodesic-radial likelihoods and chart-invariant Fisher information while maintaining isotropy, decoupling statistical modeling from numerical chart choice in manifold generative models.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis holds only inside the class of isotropic, scalar-Jacobian azimuthal charts; outside this class the impossibility result and the necessity of the RC form may not apply.","pith_extraction_headline":"Within isotropic scalar-Jacobian azimuthal charts no base distribution preserves geodesic-radial likelihoods, chart-invariant radial Fisher information, and tangent-space isotropy unless it takes the specific Radial Compensation form."},"references":{"count":29,"sample":[{"doi":"","year":2022,"title":"Heli Ben-Hamu, Samuel Cohen, Joey Bose, Brandon Amos, Maximillian Nickel, Aditya Grover, Ricky T. Q. Chen, and Yaron Lipman. Matching normalizing flows and probability paths on manifolds. In Proceedin","work_id":"a4d75e1f-2e89-4d4f-940f-153d455572d0","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Joey Bose, Ariella Smofsky, Renjie Liao, Prakash Panangaden, and William L. Hamilton. Latent variable modelling with hyperbolic normalizing flows. InProceedings of the 37th International Conference on","work_id":"471b459b-19a0-47b4-95b8-139d3c2a8ca6","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Ricky T. Q. Chen and Yaron Lipman. Flow matching on general geometries. InInternational Conference on Learning Representations (ICLR), 2024. 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