Equivariant Geodesic Networks: End-to-End Classification on the SPD Manifold

We introduce Equivariant Geodesic Networks (EGN), an end-to-end classifier that operates directly on the symmetric positive definite (SPD) manifold rather than projecting features to Euclidean space. EGN is composed of an equivariant mapping stage, Riemannian pooling, a geometric bias term, and geodesic attention layers, together with a numerically stable matrix-logarithm backward pass.

By respecting the underlying manifold geometry throughout the network, EGN improves stability and accuracy over Euclidean baselines and naive manifold approaches on covariance-based representations. The architecture is released as the open-source package egnlib.

egnlib — Equivariant Geodesic Networks for the SPD manifold 2026 • Authored and released an open-source Python package on the Python Package Index implementing the complete EGN architecture, including equivariant mapping, Riemannian pooling, geometric bias, and geodesic attention layers, with a stable matrix-logarithm backward pass. • Distributed under pip install egnlib; Link: https://pypi.org/project/egnlib/ Source: https://github.com/kraihan/EGN

Recommended citation: Md Raihan Khan and Airin Akter Tania, "Equivariant Geodesic Networks: End-to-End Classification on the SPD Manifold," AAAI 2026 (under review).