  This paper introduces a novel algorithm to approximate the matrix with
minimum nuclear norm among all matrices obeying a set of convex constraints.
This problem may be understood as the convex relaxation of a rank minimization
problem, and arises in many important applications as in the task of recovering
a large matrix from a small subset of its entries (the famous Netflix problem).
Off-the-shelf algorithms such as interior point methods are not directly
amenable to large problems of this kind with over a million unknown entries.
  This paper develops a simple first-order and easy-to-implement algorithm that
is extremely efficient at addressing problems in which the optimal solution has
low rank. The algorithm is iterative and produces a sequence of matrices (X^k,
Y^k) and at each step, mainly performs a soft-thresholding operation on the
singular values of the matrix Y^k. There are two remarkable features making
this attractive for low-rank matrix completion problems. The first is that the
soft-thresholding operation is applied to a sparse matrix; the second is that
the rank of the iterates X^k is empirically nondecreasing. Both these facts
allow the algorithm to make use of very minimal storage space and keep the
computational cost of each iteration low. We provide numerical examples in
which 1,000 by 1,000 matrices are recovered in less than a minute on a modest
desktop computer. We also demonstrate that our approach is amenable to very
large scale problems by recovering matrices of rank about 10 with nearly a
billion unknowns from just about 0.4% of their sampled entries. Our methods are
connected with linearized Bregman iterations for l1 minimization, and we
develop a framework in which one can understand these algorithms in terms of
well-known Lagrange multiplier algorithms.
