Approximated Geodesic Updates with Principal Natural Gradients

We propose a novel optimization algorithm which overcomes two drawbacks of Amari´s natural gradient updates for information geometry. First, prewhitening the tangent vectors locally converts a Riemannian manifold to an Euclidean space so that the additive parameter update sequence approximates geodesics. Second, we prove that dimensionality reduction of natural gradients is necessary for learning multidimensional linear transformations. Removal of minor components also leads to noise reduction and better computational efficiency. The proposed method demonstrates faster and more robust convergence in the simulations on recovering a Gaussian mixture of artificial data and on discriminative learning of ionosphere data.