Spectral Geometric Matrix Completion

TL;DR

This paper introduces a spectral geometric perspective to deep matrix factorization, enabling explicit regularization that leverages underlying geometric relations in matrix completion tasks, improving performance on real-world benchmarks.

Contribution

It presents a novel spectral geometric interpretation of DMF, allowing explicit regularization that exploits geometric relations, enhancing matrix completion in applications like recommender systems.

Findings

Improved matrix completion performance on real benchmarks.

Successful application of deep linear networks in this context.

Effective incorporation of geometric relations into DMF models.

Abstract

Deep Matrix Factorization (DMF) is an emerging approach to the problem of matrix completion. Recent works have established that gradient descent applied to a DMF model induces an implicit regularization on the rank of the recovered matrix. In this work we interpret the DMF model through the lens of spectral geometry. This allows us to incorporate explicit regularization without breaking the DMF structure, thus enjoying the best of both worlds. In particular, we focus on matrix completion problems with underlying geometric or topological relations between the rows and/or columns. Such relations are prevalent in matrix completion problems that arise in many applications, such as recommender systems and drug-target interaction. Our contributions enable DMF models to exploit these relations, and make them competitive on real benchmarks, while exhibiting one of the first successful applications of deep linear networks.