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The sparse factorization of a large matrix is fundamental in modernstatistical learning. In particular, the sparse singular value decompositionand its variants have been utilized in multivariate regression, factoranalysis, biclustering, vector time series modeling, among others. The appealof this factorization is owing to its power in discovering ahighly-interpretable latent association network, either between samples andvariables or between responses and predictors. However, many existing methodsare either ad hoc without a general performance guarantee, or arecomputationally intensive, rendering them unsuitable for large-scale studies.We formulate the statistical problem as a sparse factor regression and tackleit with a divide-and-conquer approach. In the first stage of division, weconsider both sequential and parallel approaches for simplifying the task intoa set of co-sparse unit-rank estimation (CURE) problems, and establish thestatistical underpinnings of these commonly-adopted and yet poorly understooddeflation methods. In the second stage of division, we innovate a contendedstagewise learning technique, consisting of a sequence of simple incrementalupdates, to efficiently trace out the whole solution paths of CURE. Ouralgorithm has a much lower computational complexity than alternating convexsearch, and the choice of the step size enables a flexible and principledtradeoff between statistical accuracy and computational efficiency. Our work isamong the first to enable stagewise learning for non-convex problems, and theidea can be applicable in many multi-convex problems. Extensive simulationstudies and an application in genetics demonstrate the effectiveness andscalability of our approach.

Statistically Guided Divide-and-Conquer for Sparse Factorization of Large Matrix