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

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.