  Low-rank matrix completion (LRMC) problems arise in a wide variety of
applications. Previous theory mainly provides conditions for completion under
missing-at-random samplings. This paper studies deterministic conditions for
completion. An incomplete $d \times N$ matrix is finitely rank-$r$ completable
if there are at most finitely many rank-$r$ matrices that agree with all its
observed entries. Finite completability is the tipping point in LRMC, as a few
additional samples of a finitely completable matrix guarantee its unique
completability. The main contribution of this paper is a deterministic sampling
condition for finite completability. We use this to also derive deterministic
sampling conditions for unique completability that can be efficiently verified.
We also show that under uniform random sampling schemes, these conditions are
satisfied with high probability if $O(\max\{r,\log d\})$ entries per column are
observed. These findings have several implications on LRMC regarding lower
bounds, sample and computational complexity, the role of coherence, adaptive
settings and the validation of any completion algorithm. We complement our
theoretical results with experiments that support our findings and motivate
future analysis of uncharted sampling regimes.
