Fast Exact Matrix Completion: A Unified Optimization Framework for Matrix Completion

TL;DR

This paper introduces fastImpute, a non-convex gradient descent-based method for matrix completion that is scalable, accurate, and significantly faster than existing methods, applicable to large matrices with or without side information.

Contribution

The paper presents fastImpute, a novel non-convex optimization framework for matrix completion that guarantees convergence to a global minimum and scales efficiently to very large matrices.

Findings

fastImpute achieves over 75% lower MAPE in high missing data scenarios

It is 15 times faster than current state-of-the-art methods

Performs competitively in accuracy and speed on synthetic and real datasets

Abstract

We formulate the problem of matrix completion with and without side information as a non-convex optimization problem. We design fastImpute based on non-convex gradient descent and show it converges to a global minimum that is guaranteed to recover closely the underlying matrix while it scales to matrices of sizes beyond $10^5 \times 10^5$. We report experiments on both synthetic and real-world datasets that show fastImpute is competitive in both the accuracy of the matrix recovered and the time needed across all cases. Furthermore, when a high number of entries are missing, fastImpute is over $75\%$ lower in MAPE and $15$ times faster than current state-of-the-art matrix completion methods in both the case with side information and without.