Generalized Fisher Kernel with Bregman Divergence

Abstract

The Fisher kernel has good statistical properties. However, from a practical point of view, the necessary distributional assumptions complicate the applicability. We approach the solution to this problem with the NMF (Non-negative Matrix Factorization) methods, which with adequate normalization conditions, provide stochastic matrices. Using the Bregman divergence as the objective function, formally equivalent solutions appear for the specific forms of the functionals involved. We show that simply by taking these results and plug-in into the general expression of the NMF kernel, obtained with purely algebraic techniques, without any assumptions about the distribution of the parameters, the properties of the Fisher kernel hold, and it is a convenient procedure to use this kernel the situations in which they are needed we derive the expression of the information matrix of Fisher. In this work, we have limited the study to the Gaussian metrics, KL (Kullback-Leibler), and I-divergence.