A Riemannian Framework for Normalized SPD Matrix Learning

Symmetric positive definite matrices are widely used to represent covariance and second-order statistics in vision, neuroscience, and time-series learning. Existing methods often rely on the Log-Euclidean metric, which applies the scalar logarithm to eigenvalues and maps SPD matrices to a Euclidean space. However, many practical SPD representations are normalized or regularized so that their spectra lie in a bounded interval, typically $0\prec X\prec I$. In this setting, the logarithm is only one possible coordinate choice, and alternative spectral geometries remain largely unexplored.We propose a unified framework for Normalized SPD learning based on diffeomorphic spectral coordinates. Given any scalar diffeomorphism $\varphi :(0,1)\to \mathbb{R}$, we lift it to SPD matrices by applying it to eigenvalues, obtaining a global coordinate map from bounded SPD matrices to symmetric matrices. This construction induces a pullback Euclidean geometry with explicit distances, geodesics, and Riemannian operators. We instantiate the framework with logit, algebraic square-root, rational, tangent, and Gumbel coordinates, which differ in symmetry, boundary behavior, and numerical conditioning.The proposed coordinates can be plugged into existing SPD architectures without changing their overall design. We evaluate them in SPD residual networks, diffeomorphic generative modeling, covariance pooling for visual recognition, and large-matrix approximation settings. Experiments across action recognition, EEG/fMRI analysis, fine-grained recognition, and ImageNet-scale covariance models show that spectral coordinate choice is an important design dimension for bounded SPD learning.