This talk is concerned with noisy matrix completion: given partial and corrupted entries of a large low-rank matrix, how to estimate and infer the underlying matrix? Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practice. However, the statistical stability guarantees of this approach is still far from optimal in the noisy setting, falling short of explaining the empirical success. Moreover, it is generally very challenging to pin down the distributions of the convex solution, which presents a major roadblock in assessing the uncertainty, or “confidence”, for the obtained estimates --- a crucial task at the core of statistical inference.  

Our recent work makes progress towards understanding stability and uncertainty quantification for matrix completion. When the rank of the unknown matrix is a constant:  (1) we demonstrate that convex programming achieves near-optimal estimation errors vis-a-vis random noise; (2) we develop a de-biased estimator that admits accurate distributional characterizations, thus enabling asymptotically optimal inference.  All of this is enabled by bridging convex relaxation with the nonconvex Burer-Monteiro approach, a seemingly distinct algorithmic paradigm that is provably robust against noise.

This is joint work with Cong Ma, Yuling Yan, Yuejie Chi, and Jianqing Fan.