  The problem of completing a low-rank matrix from a subset of its entries is
often encountered in the analysis of incomplete data sets exhibiting an
underlying factor model with applications in collaborative filtering, computer
vision and control. Most recent work had been focused on constructing efficient
algorithms for exact or approximate recovery of the missing matrix entries and
proving lower bounds for the number of known entries that guarantee a
successful recovery with high probability. A related problem from both the
mathematical and algorithmic point of view is the distance geometry problem of
realizing points in a Euclidean space from a given subset of their pairwise
distances. Rigidity theory answers basic questions regarding the uniqueness of
the realization satisfying a given partial set of distances. We observe that
basic ideas and tools of rigidity theory can be adapted to determine uniqueness
of low-rank matrix completion, where inner products play the role that
distances play in rigidity theory. This observation leads to an efficient
randomized algorithm for testing both local and global unique completion.
Crucial to our analysis is a new matrix, which we call the completion matrix,
that serves as the analogue of the rigidity matrix.
