Beyond Rigid Geometries: The Spline-Pullback Metric for Universal Diffeomorphic SPD Representation Learning

TL;DR

This paper introduces the Spline-Pullback Metric (SPM), a novel, flexible geometric framework for SPD matrix learning that overcomes limitations of fixed Riemannian metrics, enabling better expressivity and stability.

Contribution

The paper proposes SPM, a universal diffeomorphic metric parameterized by B-splines, which subsumes existing metrics and improves SPD representation learning.

Findings

Achieves state-of-the-art results on three datasets.

Provides a topologically bijective and stable spectral geometry.

Enables localized non-linear spectral modeling.

Abstract

The integration of Symmetric Positive Definite (SPD) matrices into deep learning has historically relied on fixed algebraic Riemannian metrics. Analogous to hand-crafted features in classical machine learning, these static formulations impose rigid geometries limiting network expressivity and adaptability. Recent attempts to parameterize these geometries often violate the axioms of primary matrix functions through unconstrained powers or rank-dependent scaling, inviting spatial folding, loss of global surjectivity, and gradient collapse at spectral singularities. In this paper, we introduce the Spline-Pullback Metric (SPM), instantiated as Spectral-SPM and Cholesky-SPM, marking a paradigm shift from static metric selection to universal geometric approximation. By parameterizing the global diffeomorphism via a rank-invariant, monotonically constrained B-spline, SPM acts as a dense universal approximator for strictly increasing $C^1$ diffeomorphisms and theoretically subsumes existing pullback metrics while enabling localized non-linear spectral modelling. Topologically, SPM provides a globally bijective pullback geometry precluding rank-swapping discontinuities and gradient instabilities. Empirically, SPM achieves a state-of-the-art performance across 3 datasets utilizing Linear Probes, SPDNets, and deep Riemannian ResNets.