# Low Rank Matrix Completion with Exponential Family Noise

arXiv: https://arxiv.org/abs/1502.06919 · HTML: https://mlsanity.com/paper/1502.06919/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Feb 2015.

## Built on by (most cited first, 4 of 4)

- [Color Image Inpainting via Robust Pure Quaternion Matrix Completion: Error Bound and Weighted Loss](https://mlsanity.com/paper/2202.02063/) (2022)
- [Parallel integrative learning for large-scale multi-response regression with incomplete outcomes](https://mlsanity.com/paper/2104.05076/) (2021)
- [Always Valid Risk Monitoring for Online Matrix Completion](https://mlsanity.com/paper/2211.10363/) (2022)
- [Exponential-Family Tensor Completion via Nonconvex Dual Total-Variation Regularization](https://mlsanity.com/paper/2606.30958/) (2026)

## Abstract (arXiv)

The matrix completion problem consists in reconstructing a matrix from a sample of entries, possibly observed with noise. A popular class of estimator, known as nuclear norm penalized estimators, are based on minimizing the sum of a data fitting term and a nuclear norm penalization. Here, we investigate the case where the noise distribution belongs to the exponential family and is sub-exponential. Our framework alllows for a general sampling scheme. We first consider an estimator defined as the minimizer of the sum of a log-likelihood term and a nuclear norm penalization and prove an upper bound on the Frobenius prediction risk. The rate obtained improves on previous works on matrix completion for exponential family. When the sampling distribution is known, we propose another estimator and prove an oracle inequality w.r.t. the Kullback-Leibler prediction risk, which translates immediatly into an upper bound on the Frobenius prediction risk. Finally, we show that all the rates obtained are minimax optimal up to a logarithmic factor.