Iso-Riemannian Optimization on Learned Data Manifolds

Abstract: High-dimensional data that exhibit an intrinsic low-dimensional structure are ubiquitous in machine learning and data science. While various approaches allow for learning the corresponding data manifold from finite samples, performing downstream tasks such as optimization directly on these learned manifolds presents a significant challenge. This work introduces a principled framework for optimization on learned data manifolds using iso-Riemannian geometry. Our approach addresses key limitations of classical Riemannian optimization in this setting, specifically, that the Levi-Civita connection fails to yield constant-speed geodesics, and that geodesic convexity assumptions break down under the learned pullback constructions commonly used in practice. To overcome these challenges, we propose new notions of monotonicity and Lipschitz continuity tailored to the iso-Riemannian setting and propose iso-Riemannian descent algorithms for which we provide a detailed convergence analysis. We demonstrate the practical effectiveness of those algorithms on both synthetic and real datasets, including MNIST under a learned pullback structure. Our approach yields interpretable barycentres, improved clustering, and provably efficient solutions to inverse problems, even in high-dimensional settings. These results establish that optimization under iso-Riemannian geometry can overcome distortions inherent to learned manifold mappings.