Title: Low-rank Distributional Matrix Completion

Abstract:

This paper investigates a distributional extension of the traditional matrix completion problem, where matrix entries are represented as probability distributions rather than scalar values. In this framework, only a portion of the matrix is observed, and for those observed entries, the actual distributions remain hidden. Instead, we have access to a finite number of samples drawn from these underlying distributions. To handle these distributional entries, we utilize kernel mean embeddings and define a specific concept of Tucker rank for matrices with distribution-valued components, thereby capturing their inherent low-rank characteristics. The infinite-dimensional nature of kernel embeddings introduces substantial methodological hurdles. To overcome these, we develop functional unfolding operators that connect our proposed distributional low-rank structure to the conventional Tucker rank used for finite-dimensional tensors. Leveraging this theoretical foundation, we present a new estimator designed for distributional matrix completion. We derive non-asymptotic error bounds to quantify the statistical efficiency of this estimator. Our extensive testing, which includes both synthetic datasets and a real-world case study, confirms the efficacy of the proposed approach.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC