Universal Diffeomorphic Representation Learning on SPD Manifolds via the Spline-Pullback Metric

Deep learning on Symmetric Positive Definite (SPD) manifolds has traditionally depended on predefined algebraic Riemannian metrics. Much like hand-engineered features in early machine learning, these fixed geometric structures constrain the expressive capacity and adaptability of the metrics. While recent works have attempted to learn parameterized geometries, they frequently breach foundational matrix function axioms via rank-dependent scaling or unbounded power operations. These violations result in spatial folding, a lack of global surjectivity, and gradient vanishing near spectral singularities. To overcome these limitations, we present the Spline-Pullback Metric (SPM), comprising both Spectral-SPM and Cholesky-SPM variants, marking a paradigm shift in SPD manifold learning from static metric choices to universal approximation. By modeling the global diffeomorphism with a monotonically constrained, rank-invariant B-spline, SPM functions as a dense universal approximator for strictly increasing diffeomorphisms. Our formulation theoretically generalizes existing pullback metrics while facilitating localized, non-linear modeling of the spectral domain. SPM guarantees a globally bijective pullback geometry, preventing gradient instabilities and rank-swapping discontinuities. We empirically demonstrate that SPM achieves superior performance results across five benchmark datasets utilizing Linear Probes, SPDNets, and deep Riemannian ResNet architectures.