  A new algorithm, termed subspace evolution and transfer (SET), is proposed
for solving the consistent matrix completion problem. In this setting, one is
given a subset of the entries of a low-rank matrix, and asked to find one
low-rank matrix consistent with the given observations. We show that this
problem can be solved by searching for a column space that matches the
observations. The corresponding algorithm consists of two parts -- subspace
evolution and subspace transfer. In the evolution part, we use a line search
procedure to refine the column space. However, line search is not guaranteed to
converge, as there may exist barriers along the search path that prevent the
algorithm from reaching a global optimum. To address this problem, in the
transfer part, we design mechanisms to detect barriers and transfer the
estimated column space from one side of the barrier to the another. The SET
algorithm exhibits excellent empirical performance for very low-rank matrices.
