  We describe a new algorithm, termed subspace evolution and transfer (SET),
for solving low-rank matrix completion problems. The algorithm takes as its
input a subset of entries of a low-rank matrix, and outputs one low-rank matrix
consistent with the given observations. The completion task is accomplished by
searching for a column space on the Grassmann manifold that matches the
incomplete observations. The SET algorithm consists of two parts -- subspace
evolution and subspace transfer. In the evolution part, we use a gradient
descent method on the Grassmann manifold to refine our estimate of the column
space. Since the gradient descent algorithm is not guaranteed to converge, due
to the existence of barriers along the search path, we design a new mechanism
for detecting barriers and transferring the estimated column space across the
barriers. This mechanism constitutes the core of the transfer step of the
algorithm. The SET algorithm exhibits excellent empirical performance for both
high and low sampling rate regimes.
