and Nested Wiberg Minimization
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. Recently Eriksson and
van den Hengel extended this approach to L1, minimizing ||Y - UV(U)||_1. We generalize
their approach beyond factorization to minimize an arbitrary function that is nonlinear
in each of two sets of variables. We demonstrate the idea with a practical Wiberg
algorithm for L1 bundle adjustment. 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 L1 projective
bundle adjustment, solving for camera matrices, points, and projective depths. We also
revisit L1 factorization, giving a greatly simplified presentation of Wiberg L1
factorization, and presenting a successive linear programming factorization algorithm.
Successive linear programming outperforms L1 Wiberg for most large inputs, establishing
a new state-of-the-art for for those cases.