Obtaining error-minimizing estimates and universal entry-wise error bounds for low-rank matrix completion

Motivated by the requirements of large scale data analysis, we provide means of recon- structing and denoising single entries of incomplete and noisy low-rank matrices, as opposed to the common approach in matrix completion which yields only an estimate for the whole matrix. We provide explicit algorithms for calculating an a-priori error bound for each entry, and a method to determine the entry itself - in the case of rank one, where the algebraic combinatorics is completely known; we describe how analogous algorithms for arbitrary rank can be constructed once certain algebraic questions are solved. We show on a log-noise model that the error-minimizing estimate qualitatively equals those of a state-of-the-art nuclear norm heuristic and OptSpace, and the confidence estimates predict - entry-wise - the estimation errors of those algorithms, and the single-entry reconstruction method presented.