Fast and Sample Efficient Inductive Matrix Completion via Multi-Phase Procrustes Flow

TL;DR

This paper introduces a new gradient-based algorithm for inductive matrix completion that significantly reduces sample and computational complexity, achieving linear convergence with favorable dependencies on features and ambient dimension.

Contribution

The paper proposes a novel multi-phase Procrustes flow algorithm that improves sample efficiency and convergence rate for inductive matrix completion.

Findings

Linear convergence rate achieved.

Sample complexity depends linearly on number of features.

Effective on both synthetic and real data.

Abstract

We revisit the inductive matrix completion problem that aims to recover a rank-$r$ matrix with ambient dimension $d$ given $n$ features as the side prior information. The goal is to make use of the known $n$ features to reduce sample and computational complexities. We present and analyze a new gradient-based non-convex optimization algorithm that converges to the true underlying matrix at a linear rate with sample complexity only linearly depending on $n$ and logarithmically depending on $d$. To the best of our knowledge, all previous algorithms either have a quadratic dependency on the number of features in sample complexity or a sub-linear computational convergence rate. In addition, we provide experiments on both synthetic and real world data to demonstrate the effectiveness of our proposed algorithm.