ICML 2025

Score-based Pullback Riemannian Geometry:
Extracting the Data Manifold Geometry using Anisotropic Flows
Willem Diepeveen * 1 Georgios Batzolis * 2 Zakhar Shumaylov 3 Carola-Bibiane Sch¨onlieb 3
Abstract
Data-driven Riemannian geometry has emerged
as a powerful tool for interpretable representation
learning, offering improved efficiency in down-
stream tasks. Moving forward, it is crucial to
balance cheap manifold mappings with efficient
training algorithms. In this work, we integrate
concepts from pullback Riemannian geometry
and generative models to propose a framework
for data-driven Riemannian geometry that is scal-
able in both geometry and learning: score-based
pullback Riemannian geometry. Focusing on uni-
modal distributions as a first step, we propose
a score-based Riemannian structure with closed-
form geodesics that pass through the data prob-
ability density. With this structure, we construct
a Riemannian autoencoder (RAE) with error
bounds for discovering the correct data manifold
dimension. This framework can naturally be used
with anisotropic normalizing flows by adopting
isometry regularization during training. Through
numerical experiments on diverse datasets, includ-
ing image data, we demonstrate that the proposed
framework produces high-quality geodesics pass-
ing through the data support, reliably estimates
the intrinsic dimension of the data manifold, and
provides a global chart of the manifold. To the
best of our knowledge, this is the first scalable
framework for extracting the complete geometry
of the data manifold.
*Equal contribution
1Department of Mathematics, Univer-
sity of California, Los Angeles, USA 2Department of Engi-
neering, University of Cambridge, UK 3Faculty of Mathemat-
ics, University of Cambridge, UK. Correspondence to: Willem
Diepeveen <wdiepeveen@math.ucla.edu>, Georgios Batzolis
<gb511@cam.ac.uk>.
Proceedings of the 42 nd International Conference on Machine
Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025
by th...