🤖 AI Summary
This work addresses the instability and lack of convergence guarantees in training neural networks on Riemannian manifolds, which arise from neglecting manifold curvature. To this end, the authors propose a curvature-aware training framework that replaces Euclidean derivative supervision with a covariant Sobolev loss, aligns gradients via parallel transport, and incorporates a Laplace–Beltrami regularizer to enhance stability. For the first time, the framework explicitly integrates manifold curvature and the Jacobian of parallel transport into Sobolev-based training, enabling the derivation of geometry-dependent smoothness constants. Building on this, the authors establish curvature-aware linear and quadratic convergence theories. Empirical validation on Lie groups such as SO(3) and SE(3), as well as tasks involving surface imaging and physics-informed learning, demonstrates stable training dynamics and confirms the predicted theoretical convergence rates.
📝 Abstract
We introduce Manifold Sobolev Informed Neural Optimization (MSINO), a curvature aware training framework for neural networks defined on Riemannian manifolds. The method replaces standard Euclidean derivative supervision with a covariant Sobolev loss that aligns gradients using parallel transport and improves stability via a Laplace Beltrami smoothness regularization term.
Building on classical results in Riemannian optimization and Sobolev theory on manifolds, we derive geometry dependent constants that yield (i) a Descent Lemma with a manifold Sobolev smoothness constant, (ii) a Sobolev Polyak Lojasiewicz inequality giving linear convergence guarantees for Riemannian gradient descent and stochastic gradient descent under explicit step size bounds, and (iii) a two step Newton Sobolev method with local quadratic contraction in curvature controlled neighborhoods.
Unlike prior Sobolev training in Euclidean space, MSINO provides training time guarantees that explicitly track curvature and transported Jacobians. Applications include surface imaging, physics informed learning settings, and robotics on Lie groups such as SO(3) and SE(3). The framework unifies value and gradient based learning with curvature aware convergence guarantees for neural training on manifolds.