  We present a novel algebraic combinatorial view on low-rank matrix completion
based on studying relations between a few entries with tools from algebraic
geometry and matroid theory. The intrinsic locality of the approach allows for
the treatment of single entries in a closed theoretical and practical
framework. More specifically, apart from introducing an algebraic combinatorial
theory of low-rank matrix completion, we present probability-one algorithms to
decide whether a particular entry of the matrix can be completed. We also
describe methods to complete that entry from a few others, and to estimate the
error which is incurred by any method completing that entry. Furthermore, we
show how known results on matrix completion and their sampling assumptions can
be related to our new perspective and interpreted in terms of a completability
phase transition.
