Riemannian structure of some new gradient descent learning algorithms

We consider some generalizations of the classical LMS learning algorithm including the exponentiated gradient (EG) algorithm. We show how one can develop these algorithms in terms of a prior distribution over the weight space. Our framework subsumes the notion of "link-functions". Differential geometric methods are used to develop the algorithms as gradient descent with respect to the natural gradient in the Riemannian structure induced by the prior distribution. This provides a Bayesian Riemannian interpretation of the EG and related algorithms. We relate our work to that of Amari (1985, 1997, 1998) and others who used similar tools in a different manner. Simulation experiments illustrating the behaviour of the new algorithms are presented.