Practical low-rank matrix approximation under robust L<inf>1</inf>-norm

Abstract

A great variety of computer vision tasks, such as rigid/nonrigid structure from motion and photometric stereo, can be unified into the problem of approximating a low-rank data matrix in the presence of missing data and outliers. To improve robustness, the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm measurement has long been recommended. Unfortunately, existing methods usually fail to minimize the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -based nonconvex objective function sufficiently. In this work, we propose to add a convex trace-norm regularization term to improve convergence, without introducing too much heterogenous information. We also customize a scalable first-order optimization algorithm to solve the regularized formulation on the basis of the augmented Lagrange multiplier (ALM) method. Extensive experimental results verify that our regularized formulation is reasonable, and the solving algorithm is very efficient, insensitive to initialization and robust to high percentage of missing data and/or outliers <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> .