Accelerated Gradient and Block-Wise Gradient Methods for Big Data Factorization

A problem often encountered in analysis of the large-scale data pertains to approximation of a given matrix \(A\in \mathbf {R}^{m\times n}\) by \(UV^T\), where \(U\in \mathbf {R}^{m\times r}\), \(V\in \mathbf {R}^{n\times r}\) and \(r < \min \{ m, n \}\). The aim of this paper is to tackle this problem through proposing an accelerated gradient descent algorithm as well as its stochastic counterpart. These frameworks are suitable candidates to surmount the computational difficulties in computing the SVD form of big matrices. On the other hand, big data are usually presented and stored in some fixed-size blocks, which is an incentive to further propose a block-wise gradient descent algorithm for their low-rank approximation. A stochastic block-wise gradient method will further be suggested to enhance the computational efficiencies when a large number of blocks are presented in the problem. Under some standard assumptions, we investigate the convergence property of the block-wise approach. Computational results for both synthetic data as well as the real-world data are provided in this paper.