How Geometry Shapes Generative Model Evaluation on Riemannian Manifolds

Abstract

On Riemannian manifolds, generative-model evaluation is geometry-dependent since curvature and dimension jointly determine distances and volume growth. Our framework systematically compares geodesic, ambient, and tangent-space versions of Wasserstein and MMD across different geometries. Using controlled-noise perturbations and Riemannian Flow Matching samples, we study metric agreement, noise sensitivity, and their relation to geometry-induced volume scaling. We find that no metric is uniformly reliable: evaluation distance can affect both measured discrepancies and noise sensitivity, varying across curvature and dimension. Our analysis leads to practical recommendations for reporting evaluation distances, comparing metric types, and calibrating metrics to the geometry of the samples.

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BibTeX

@misc{oberski2026geometry,
  title = {{How Geometry Shapes Generative Model Evaluation on Riemannian Manifolds}},
  author = {Misha Oberski and Valentijn Oldenburg and Xanthos Kekkos and Soham Chaudhuri and Emma Oosterhuis and Floris de Kam and Sagi Schwartz and Alison Pouplin and Federica Valeau},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33731},
  eprint = {2610.33731},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33731},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - How Geometry Shapes Generative Model Evaluation on Riemannian Manifolds
AU  - Misha Oberski
AU  - Valentijn Oldenburg
AU  - Xanthos Kekkos
AU  - Soham Chaudhuri
AU  - Emma Oosterhuis
AU  - Floris de Kam
AU  - Sagi Schwartz
AU  - Alison Pouplin
AU  - Federica Valeau
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33731
UR  - https://arxiv.org/abs/2610.33731
AB  - On Riemannian manifolds, generative-model evaluation is geometry-dependent since curvature and dimension jointly determine distances and volume growth. Our framework systematically compares geodesic, ambient, and tangent-space versions of Wasserstein and MMD across different geometries. Using controlled-noise perturbations and Riemannian Flow Matching samples, we study metric agreement, noise sensitivity, and their relation to geometry-induced volume scaling. We find that no metric is uniformly reliable: evaluation distance can affect both measured discrepancies and noise sensitivity, varying across curvature and dimension. Our analysis leads to practical recommendations for reporting evaluation distances, comparing metric types, and calibrating metrics to the geometry of the samples.
ER  -

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