Inversion-Free Natural Gradient Descent on Riemannian Manifolds

The natural gradient method is a central tool for statistical optimisation, but its broader application is hindered by the assumption of a Euclidean parameter space, the repeated estimation of the Fisher information matrix (FIM), and the computational cost of its subsequent inversion. This paper proposes an intrinsic, inversion-free natural gradient method for statistical models whose parameters lie on general Riemannian manifolds. Formulating statistical optimisation in this non-Euclidean setting allows for the natural enforcement of parameter constraints, the elimination of non-identifiable parameters, and the exploitation of geodesic convexity. Our algorithm is based on a moving approximation of the inverse FIM, which is maintained directly on the manifold. This approximation is efficiently updated with new score vectors using low-rank matrix identities. We prove almost-sure convergence rates of O(log s / sα) for the sequence of iterates, and a similar rate for the approximate FIM. A limited-memory variant with sub-quadratic storage complexity is further proposed for large-scale applications. We demonstrate the efficacy of our method on variational Bayes within the Bures-Wasserstein manifold, normalising flows on the Stiefel manifold, and reduced-rank logistic regression.