Low-rank Distributional Matrix Completion
For practitioners dealing with distribution-valued data (e.g., in finance or neuroscience), this provides a principled completion method, though the contribution is incremental as it extends existing matrix completion to a distributional setting.
This paper tackles distributional matrix completion, where entries are probability distributions observed via samples. They propose a low-rank estimator using kernel mean embeddings and Tucker rank, achieving non-asymptotic error bounds and showing effectiveness on synthetic and real data.
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.