General and Nested Wiberg Minimization: L 2 and Maximum Likelihood

Abstract

Wiberg matrix factorization breaks a matrix Y into low-rank factors U and V by solving for V in closed form given U , linearizing V ( U ) about U , and iteratively minimizing || Y  −  UV ( U )||_2 with respect to U only. This approach factors the matrix while effectively removing V from the minimization. We generalize the Wiberg approach beyond factorization to minimize an arbitrary function that is nonlinear in each of two sets of variables. In this paper we focus on the case of L _2 minimization and maximum likelihood estimation (MLE), presenting an L _2 Wiberg bundle adjustment algorithm and a Wiberg MLE algorithm for Poisson matrix factorization. We also show that one Wiberg minimization can be nested inside another, effectively removing two of three sets of variables from a minimization. We demonstrate this idea with a nested Wiberg algorithm for L _2 projective bundle adjustment, solving for camera matrices, points, and projective depths.