General and Nested Wiberg Minimization

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.

• Machine Perception