  Low-rank modeling plays a pivotal role in signal processing and machine
learning, with applications ranging from collaborative filtering, video
surveillance, medical imaging, to dimensionality reduction and adaptive
filtering. Many modern high-dimensional data and interactions thereof can be
modeled as lying approximately in a low-dimensional subspace or manifold,
possibly with additional structures, and its proper exploitations lead to
significant reduction of costs in sensing, computation and storage. In recent
years, there is a plethora of progress in understanding how to exploit low-rank
structures using computationally efficient procedures in a provable manner,
including both convex and nonconvex approaches. On one side, convex relaxations
such as nuclear norm minimization often lead to statistically optimal
procedures for estimating low-rank matrices, where first-order methods are
developed to address the computational challenges; on the other side, there is
emerging evidence that properly designed nonconvex procedures, such as
projected gradient descent, often provide globally optimal solutions with a
much lower computational cost in many problems. This survey article will
provide a unified overview of these recent advances on low-rank matrix
estimation from incomplete measurements. Attention is paid to rigorous
characterization of the performance of these algorithms, and to problems where
the low-rank matrix have additional structural properties that require new
algorithmic designs and theoretical analysis.
