From Wirtinger to Fisher Information Inequalities on Spheres and Rotation Groups

Abstract

The concepts of Fisher Information matrix and covariance are generalized to the setting of probability densities on spheres and rotation groups, and inequalities relating these quantities are derived. Probability density functions on these spaces arise in various scenarios in the fields of structural biology, robotics, and computer vision. The approach taken is to first derive matrix generalizations of Wirtinger's inequality for tori and spheres and generalize these to rotation groups. Then new inequalities are derived that relate the covariances of probability density functions on spheres and rotation groups with their Fisher information. These inequalities are different than the Cramér-Rao bound, and can be used to estimate the rate of increase of the entropy of a diffusion process.