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)∥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.