  We propose a general framework for reconstructing and denoising single
entries of incomplete and noisy entries. We describe: effective algorithms for
deciding if and entry can be reconstructed and, if so, for reconstructing and
denoising it; and a priori bounds on the error of each entry, individually. In
the noiseless case our algorithm is exact. For rank-one matrices, the new
algorithm is fast, admits a highly-parallel implementation, and produces an
error minimizing estimate that is qualitatively close to our theoretical and
the state-of-the-are Nuclear Norm and OptSpace methods.
