Riemannian Integrated Gradients: A Geometric View of Explainable AI
Federico Costanza, Lachlan Simpson
TL;DR
This work addresses explainability for neural networks on non-Euclidean data by extending gradient-based attributions to Riemannian manifolds. It introduces Riemannian Integrated Gradients (RIG) as a manifold-consistent generalisation of Integrated Gradients (IG), using a bilinear path-attribution form and geodesic connections between points. A key contribution is showing that, with a natural choice of basis, RIG attributions correspond to the eigenvalues of a symmetric endomorphism $Q_{F,\gamma}(p)$ associated with the path attribution form, and that RIG reduces to IG in Euclidean space. The framework preserves isometry-invariance like properties and provides a geometrically meaningful, basis-dependent attribution mechanism that respects manifold structure, with planned empirical validation on geometries beyond Euclidean spaces.
Abstract
We introduce Riemannian Integrated Gradients (RIG); an extension of Integrated Gradients (IG) to Riemannian manifolds. We demonstrate that RIG restricts to IG when the Riemannian manifold is Euclidean space. We show that feature attribution can be phrased as an eigenvalue problem where attributions correspond to eigenvalues of a symmetric endomorphism.