Abstract
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.
Highlights
M ODERN information processing and machine learning often have to deal with low-rank matrix factorization
Several problems provably enjoy benign optimization landscape when the sample size is sufficiently large, in the sense that there is no spurious local minima, i.e. all local minima are global minima, and that the only undesired stationary points are strict saddle points [28]–[32]. These important messages inspire a recent flurry of activities in trheTdweosi-gSntaogfetwAopcpornotarcahst.inMg oatligvoartietdhmbiyc approaches: the existence of a basin of attraction, a large number of works follow a two-stage paradigm: (1) initialization, which locates an initial guess within the basin; (2) iterative refinement, which successively refines the estimate without leaving the basin
This problem stems from interpreting principal component analysis (PCA) from an optimization perspective, which has a long history in the literature of neural networks and unsupervised learning; see for example [36]–[41]
Summary
M ODERN information processing and machine learning often have to deal with (structured) low-rank matrix factorization. Date of publication August 23, 2019; date of current version September 16, 2019. A common goal of these problems is to develop reliable, scalable, and robust algorithms to estimate a low-rank matrix of interest, from potentially noisy, nonlinear, and highly incomplete observations
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