Uncertainty Quantification For Low-Rank Matrix Completion With Heterogeneous and Sub-Exponential Noise

TL;DR

This paper develops a method to quantify uncertainty in low-rank matrix completion when data is noisy and heterogeneously distributed, extending previous Gaussian-based models to more realistic noise types like Poisson and Binary.

Contribution

It characterizes the distribution of estimated matrix entries under heterogeneous sub-exponential noise, providing explicit formulas for Poisson and Binary noise models.

Findings

Distribution of estimates is characterized under heterogenous sub-exponential noise.

Explicit formulas derived for Poisson and Binary distributed observations.

Enhances uncertainty quantification in practical matrix completion applications.

Abstract

The problem of low-rank matrix completion with heterogeneous and sub-exponential (as opposed to homogeneous and 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.