Dictionary Learning for Massive Matrix Factorization

Sparse matrix factorization is a popular tool to obtain interpretable data decompositions, which are also effective to perform data completion or denoising. Its applicability to large datasets has been addressed with online and randomized methods, that reduce the complexity in one of the matrix dimension, but not in both of them. In this paper, we tackle very large matrices in both dimensions. We propose a new factorization method that scales gracefully to terabyte-scale datasets. Those could not be processed by previous algorithms in a reasonable amount of time. We demonstrate the efficiency of our approach on massive functional Magnetic Resonance Imaging (fMRI) data, and on matrix completion problems for recommender systems, where we obtain significant speedups compared to state-of-the art coordinate descent methods. Matrix factorization is a flexible tool for uncovering latent factors in low-rank or sparse models. For instance, building on low-rank structure, it has proven very powerful for matrix completion, e.g. in recommender systems (Srebro et al., 2004; Candès & Recht, 2009). In signal processing and computer vision, matrix factorization with a sparse regularization is often called dictionary learning and has proven very effective for denoising and visual feature encoding (see Mairal, 2014, for a review).