Unifying Nuclear Norm and Bilinear Factorization Approaches for Low-Rank Matrix Decomposition

Ricardo Cabral, Fernando De La Torre, Joao P. Costeira, Alexandre Bernardino; Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2013, pp. 2488-2495

Low rank models have been widely used for the representation of shape, appearance or motion in computer vision problems. Traditional approaches to fit low rank models make use of an explicit bilinear factorization. These approaches benefit from fast numerical methods for optimization and easy kernelization. However, they suffer from serious local minima problems depending on the loss function and the amount/type of missing data. Recently, these lowrank models have alternatively been formulated as convex problems using the nuclear norm regularizer; unlike factorization methods, their numerical solvers are slow and it is unclear how to kernelize them or to impose a rank a priori. This paper proposes a unified approach to bilinear factorization and nuclear norm regularization, that inherits the benefits of both. We analyze the conditions under which these approaches are equivalent. Moreover, based on this analysis, we propose a new optimization algorithm and a "rank continuation" strategy that outperform state-of-theart approaches for Robust PCA, Structure from Motion and Photometric Stereo with outliers and missing data.