Uncertainty Quantification for Low-Rank Matrix Completion with Heterogeneous and Sub-Exponential Noise

The problem of low-rank matrix completion with heterogeneous and sub-exponential (as opposed to homogeneous Gaussian) noise is particularly relevant to a number of applications in modern commerce. Examples include panel sales data and data collected from web-commerce systems such as recommendation engines. An important unresolved question for this problem is characterizing the distribution of estimated matrix entries under common low-rank estimators. Such a characterization is essential to any application that requires quantification of uncertainty in these estimates and has heretofore only been available under the assumption of homogenous Gaussian noise. Here we characterize the distribution of estimated matrix entries when the observation noise is heterogeneous sub-Exponential and provide, as an application, explicit formulas for this distribution when observed entries are Poisson or Binary distributed.