  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.
