# Low-rank Distributional Matrix Completion

arXiv: https://arxiv.org/abs/2606.04176 · HTML: https://mlsanity.com/paper/2606.04176/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 43 references that are in the map.
- First posted on arXiv 2 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery](https://mlsanity.com/paper/1307.5870/) (2013)
- [A Nonconvex Low-Rank Tensor Completion Model for Spatiotemporal Traffic Data Imputation](https://mlsanity.com/paper/2003.10271/) (2020)
- [High Dimensional Statistical Estimation under Uniformly Dithered One-bit Quantization](https://mlsanity.com/paper/2202.13157/) (2022)
- [Guaranteed Functional Tensor Singular Value Decomposition](https://mlsanity.com/paper/2108.04201/) (2021)
- [Low-Rank Covariance Function Estimation for Multidimensional Functional Data](https://mlsanity.com/paper/2008.12919/) (2020)
- [Top-N Recommender System via Matrix Completion](https://mlsanity.com/paper/1601.04800/) (2016)
- [Bayesian Uncertainty Quantification for Low-Rank Matrix Completion](https://mlsanity.com/paper/2101.01299/) (2021)
- [Matrix Completion with Model-free Weighting](https://mlsanity.com/paper/2106.05850/) (2021)
- [Tensor Decomposition Meets RKHS: Efficient Algorithms for Smooth and Misaligned Data](https://mlsanity.com/paper/2408.05677/) (2024)
- [Distributional Matrix Completion via Nearest Neighbors in the Wasserstein Space](https://mlsanity.com/paper/2410.13112/) (2024)

## Abstract (arXiv)

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.