From Ricci Curvature to Metric Matching: A Simplified Approach to Geometric Transfer Learning

Keywords: Differential Geometry, Geometrical Deeplearning, Transfer Learning

TL;DR: Faster, more efficient molecular property prediction by aligning latent geometries without expensive curvature computations.

Abstract:

Riemannian geometry offers a principled frame-
work for modeling the smooth, curved latent
spaces induced by deep learning models, and has
shown particular promise in transfer learning. Re-
cent approaches improve prediction performance
by aligning the Ricci scalar curvature between
target and source domains, thereby matching their
intrinsic geometric structures. However, Ricci
curvature computation is mathematically complex
and computationally expensive, limiting its repro-
ducibility. In this work, we propose a simplified
metric matching approach that reduces both the
algorithmic complexity and computational cost.
Our method leverages an exact diffeomorphism
between target and source spaces, enabling coor-
dinate frame alignment such that, by definition
of the curvature tensor, the curvature remains un-
changed. Extensive experiments demonstrate that
our approach improves training speed by 91.1%
and reduces memory usage by 62.6% compared
to GEAR, a Ricci curvature–based method, while
maintaining comparable predictive performance.

Submission Number: 6