General and nested Wiberg minimization: L2 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 jjY UV (U)jj2 with respect to U only. This
approach factors the matrix while e ectively 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 L2 minimization
and maximum likelihood estimation (MLE), presenting an
L2 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, e ectively removing two of three sets of variables
from a minimization. We demonstrate this idea with a
nested Wiberg algorithm for L2 projective bundle
adjustment, solving for camera matrices, points, and
projective depths.
low-rank factors U and V by solving for V in closed
form given U, linearizing V (U) about U, and iteratively
minimizing jjY UV (U)jj2 with respect to U only. This
approach factors the matrix while e ectively 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 L2 minimization
and maximum likelihood estimation (MLE), presenting an
L2 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, e ectively removing two of three sets of variables
from a minimization. We demonstrate this idea with a
nested Wiberg algorithm for L2 projective bundle
adjustment, solving for camera matrices, points, and
projective depths.
Research Areas
