Spectral dimensionality reduction for Bregman information

Abstract

A spectral dimensionality reduction in dually flat spaces is formulated. On the basis of the expansion with respect to the affine coordinate, it is shown that some inner products compose Bregman divergence. Since the centroids are determined in two ways, depending on coordinate used for centering, Bregman information is also evaluated in two ways. In both cases, Bregman information is shown to be represented as a weighted sum of the eigenvalues of two inner product matrices. For given dissimilarities, the problem which finds the values of the inner product matrices is described as a semidefinite programming. As an example, numerical calculations are also presented.