Recent Developments on Matrix Completion

Abstract

The problem of Matrix Completion has been widely studied over the past decade. However, the vast majority of the solutions have proposed computationally feasible estimators with strong statistical guarantees for the case where the underlying distribution of data in the matrix is continuous. A few recent approaches have extended using similar ideas these estimators to the case where the underlying distributions belong to the exponential family. Most of these approaches assume that there is only one underlying distribution and the low rank constraint is regularized by the matrix Schatten Norm. We propose as a main result in this thesis a computationally feasible statistical approach with strong recovery guarantees along with an algorithmic framework suited for parallelization to recover a low rank matrix with partially observed entries for mixed data types in one step. We also provide extensive simulation evidence that corroborate our theoretical results. We will also include in this thesis that a comprehensive survey of the previous works matrix completion works in a unifying notation as the starting chapter. Besides the main Robust Matrix Completion algorithmic framework, we also introduce a generalized chaining algorithm, inspired by the Chained Equation methods, to enhance recovery performance on almost all matrix completion algorithms that is based on nuclear norm relaxation and regularization or alternating minimization. Comprehensive simulation results on simulated data sets will also be provided to corroborate our results. Finally, in the last chapter, we will give a brief introduction the package MatrixCompletion.jl, which contains all implementations of our proposed robust matrix completion framework.