Low-rank Distributional Matrix Completion

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.

Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)

arXiv:2606.04176 [cs.LG]

(or arXiv:2606.04176v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.04176 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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