General 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)∥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 L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , minimizing ∥Y-UV (U)∥ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> . 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 L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> 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 L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> projective bundle adjustment, solving for camera matrices, points, and projective depths. We also revisit L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> factorization, giving a greatly simplified presentation of Wiberg L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> factorization, and presenting a successive linear programming factorization algorithm. Successive linear programming outperforms L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> Wiberg for most large inputs, establishing a new state-of-the-art for for those cases.