Generalization Bounds for Inductive Matrix Completion in Low-noise Settings

TL;DR

This paper establishes new generalization bounds for inductive matrix completion under low-noise conditions, demonstrating convergence properties and dependence on noise and sample size, with implications for exact and approximate recovery.

Contribution

It provides the first generalization bounds for inductive matrix completion that scale with noise and sample size, using a novel combination of theoretical techniques.

Findings

Bounds scale with noise standard deviation and approach zero in exact recovery

Generalization bounds converge to zero as sample size increases

Logarithmic dependence on matrix size for fixed side information dimension

Abstract

We study inductive matrix completion (matrix completion with side information) under an i.i.d. subgaussian noise assumption at a low noise regime, with uniform sampling of the entries. We obtain for the first time generalization bounds with the following three properties: (1) they scale like the standard deviation of the noise and in particular approach zero in the exact recovery case; (2) even in the presence of noise, they converge to zero when the sample size approaches infinity; and (3) for a fixed dimension of the side information, they only have a logarithmic dependence on the size of the matrix. Differently from many works in approximate recovery, we present results both for bounded Lipschitz losses and for the absolute loss, with the latter relying on Talagrand-type inequalities. The proofs create a bridge between two approaches to the theoretical analysis of matrix completion, since they consist in a combination of techniques from both the exact recovery literature and the approximate recovery literature.