# Low-rank matrix completion and denoising under Poisson noise

arXiv: https://arxiv.org/abs/1907.05325 · HTML: https://mlsanity.com/paper/1907.05325/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2019.

## Built on by (most cited first, 6 of 6)

- [Spectral Entry-wise Matrix Estimation for Low-Rank Reinforcement Learning](https://mlsanity.com/paper/2310.06793/) (2023)
- [Projection-Free Algorithm for Stochastic Bi-level Optimization](https://mlsanity.com/paper/2110.11721/) (2021)
- [Robust Matrix Completion with Heavy-tailed Noise](https://mlsanity.com/paper/2206.04276/) (2022)
- [Uncertainty Quantification For Low-Rank Matrix Completion With Heterogeneous and Sub-Exponential Noise](https://mlsanity.com/paper/2110.12046/) (2021)
- [Computationally efficient reductions between some statistical models](https://mlsanity.com/paper/2402.07717/) (2024)
- [High Rank Matrix Completion via Grassmannian Proxy Fusion](https://mlsanity.com/paper/2602.02565/) (2026)

## Abstract (arXiv)

This paper considers the problem of estimating a low-rank matrix from the observation of all or a subset of its entries in the presence of Poisson noise. When we observe all entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyze several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares, and a nonconvex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed, and maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.