Categorical Matrix Completion

TL;DR

This paper introduces a method for completing matrices with categorical entries by extending one-bit matrix completion, providing theoretical bounds and demonstrating improved performance over traditional methods on real data.

Contribution

It develops a novel categorical matrix completion framework with theoretical error bounds and practical advantages over existing methods.

Findings

Theoretical bounds on recovery error are established, tight up to a factor involving the number of categories.

The method's error bounds depend on the smoothness of link functions and the number of categories.

Empirical results on MovieLens dataset show the proposed method outperforms conventional matrix completion.

Abstract

We consider the problem of completing a matrix with categorical-valued entries from partial observations. This is achieved by extending the formulation and theory of one-bit matrix completion. We recover a low-rank matrix $X$ by maximizing the likelihood ratio with a constraint on the nuclear norm of $X$, and the observations are mapped from entries of $X$ through multiple link functions. We establish theoretical upper and lower bounds on the recovery error, which meet up to a constant factor $\mathcal{O}(K^{3/2})$ where $K$ is the fixed number of categories. The upper bound in our case depends on the number of categories implicitly through a maximization of terms that involve the smoothness of the link functions. In contrast to one-bit matrix completion, our bounds for categorical matrix completion are optimal up to a factor on the order of the square root of the number of categories, which is consistent with an intuition that the problem becomes harder when the number of categories increases. By comparing the performance of our method with the conventional matrix completion method on the MovieLens dataset, we demonstrate the advantage of our method.