An Efficient Proximal Block Coordinate Homotopy Method for Large-Scale Sparse Least Squares Problems

Abstract

In this paper, an efficient and robust algorithm framework is presented for large-scale sparse least squares problems. This framework decomposes the original sparse least squares problem into a sequence of small-scale $l_1$-minimization subproblems. Every subproblem is solved by an improved $l_1$-homotopy method which differs from the original $l_1$-homotopy method by adopting a warm-start procedure and an $\varepsilon$-precision verification-correction technique. Moreover, based on a carefully designed block coordinate update strategy, the algorithm framework is proved to converge to a $\tilde \tau$-``precise" solution in a finite number of steps and the value of the objective function linearly converges. Numerical comparisons between the presented algorithms and a number of state-of-the-art algorithms on real and randomly generated data sets demonstrate the robustness and high performance of the presented algorithms. As an example, the presented algorithm only needs 13 seconds to solve an $l_1$-minimization problem with tens of millions of samples and features.