Inductive Matrix Completion: No Bad Local Minima and a Fast Algorithm

TL;DR

This paper proves that the IMC problem has no bad local minima, introduces a rank estimation scheme, and proposes GNIMC, a Gauss-Newton based method with strong theoretical guarantees and faster empirical recovery.

Contribution

It establishes the absence of bad local minima in IMC, develops a rank estimation method, and introduces GNIMC with improved convergence and recovery guarantees.

Findings

GNIMC recovers matrices faster than competing methods.

Theoretical guarantees include quadratic convergence and fewer observed entries.

No bad local minima exist in the IMC optimization landscape.

Abstract

The inductive matrix completion (IMC) problem is to recover a low rank matrix from few observed entries while incorporating prior knowledge about its row and column subspaces. In this work, we make three contributions to the IMC problem: (i) we prove that under suitable conditions, the IMC optimization landscape has no bad local minima; (ii) we derive a simple scheme with theoretical guarantees to estimate the rank of the unknown matrix; and (iii) we propose GNIMC, a simple Gauss-Newton based method to solve the IMC problem, analyze its runtime and derive recovery guarantees for it. The guarantees for GNIMC are sharper in several aspects than those available for other methods, including a quadratic convergence rate, fewer required observed entries and stability to errors or deviations from low-rank. Empirically, given entries observed uniformly at random, GNIMC recovers the underlying matrix substantially faster than several competing methods.