Low-rank Distributional Matrix Completion

2026-06-02

AI Summary

This work addresses the challenge of matrix completion when entries are probability distributions observed only indirectly through finite samples. It pioneers the extension of low-rank matrix completion to the distribution-valued setting by representing distributional entries via kernel mean embeddings. The authors introduce a Tucker rank notion for distribution-valued matrices and establish its connection to the classical finite-dimensional tensor Tucker rank through a functional expansion operator, thereby enabling a computable completion estimator. This approach effectively circumvents the theoretical and computational difficulties arising from the infinite-dimensional nature of kernel embeddings. The study derives non-asymptotic error bounds for the proposed estimator and demonstrates its efficacy through experiments on both synthetic data and real-world applications.

Abstract

We study a distributional generalization of the matrix completion problem in which each entry of the target matrix is a probability distribution rather than a scalar. In this setting, only a subset of matrix entries is observed, and even for observed entries, the underlying distributions are not directly accessible; instead, we observe finitely many samples drawn from them. To represent distributional entries, we employ kernel mean embeddings and introduce a notion of Tucker rank for distribution-valued matrices to capture their low-rank structure. The infinite-dimensional nature of kernel embeddings poses significant methodological challenges. To address this, we introduce functional unfolding operators that link the proposed distributional low-rank structure to the classical Tucker rank for finite-dimensional tensors. Based on this framework, we propose a novel estimator for distributional matrix completion. We establish non-asymptotic error bounds that characterize the statistical performance of the estimator. Extensive experiments on synthetic data and a real-world application demonstrate the effectiveness of the proposed method.

Problem

• distributional matrix completion
• probability distributions
• matrix completion
• low-rank structure
• kernel mean embeddings

Innovation

• distributional matrix completion
• kernel mean embedding
• Tucker rank
• functional unfolding
• non-asymptotic error bounds