Generalized Matrix-fractional Functions and Their Applications

Abstract

The support function of a closed convex set is a central object in convex geometry as it completely identifies the underlying set. For a particular class of sets -- the graph of matrix valued mapping $Y\mapsto -\half YY^T$ over an affine manifold $\set{Y\in\Rnm}{AY=B}$, their support functions are named generalized matrix-fractional (GMF) functions, and were first introduced by Burke and Hoheisel as a tool for unifying a wide range of applications including variational properties of linear constrained quadratic optimization problems, generalized Ky Fan norms, variational Gram functions (VGF), the Aitken's theorem and Gauss-Markov theorem in statistical estimation, and many topics in machine learning such as K-means clustering, support vector machines and multi-task learning. In the first part of the thesis we study the convex geometry of GMF functions and dramatically simplify their original representations. Second part of the thesis is devoted to the study of partial infimal projections of the sum of GMF functions and various classes of convex functions, where most applications arise.