# Uncertainty Quantification For Low-Rank Matrix Completion With Heterogeneous and Sub-Exponential Noise

arXiv: https://arxiv.org/abs/2110.12046 · HTML: https://mlsanity.com/paper/2110.12046/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 60 references that are in the map.
- First posted on arXiv 22 Oct 2021.

## Builds on (most cited first, 15 of 15)

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Matrix Completion has No Spurious Local Minimum](https://mlsanity.com/paper/1605.07272/) (2016)
- [Guaranteed Matrix Completion via Non-convex Factorization](https://mlsanity.com/paper/1411.8003/) (2014)
- [Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution](https://mlsanity.com/paper/1711.10467/) (2017)
- [Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization](https://mlsanity.com/paper/1902.07698/) (2019)
- [Inference and Uncertainty Quantification for Noisy Matrix Completion](https://mlsanity.com/paper/1906.04159/) (2019)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [Low-Rank Matrix Recovery with Scaled Subgradient Methods: Fast and Robust Convergence Without the Condition Number](https://mlsanity.com/paper/2010.13364/) (2020)
- [The Global Optimization Geometry of Low-Rank Matrix Optimization](https://mlsanity.com/paper/1703.01256/) (2017)
- [Statistical Inferences of Linear Forms for Noisy Matrix Completion](https://mlsanity.com/paper/1909.00116/) (2019)
- [Low-rank matrix completion and denoising under Poisson noise](https://mlsanity.com/paper/1907.05325/) (2019)
- [Learning Treatment Effects in Panels with General Intervention Patterns](https://mlsanity.com/paper/2106.02780/) (2021)
- [Deep density networks and uncertainty in recommender systems](https://mlsanity.com/paper/1711.02487/) (2017)
- [Bayesian Matrix Completion Approach to Causal Inference with Panel Data](https://mlsanity.com/paper/1911.01287/) (2019)

## Built on by (most cited first, 1 of 1)

- [Speed Up the Cold-Start Learning in Two-Sided Bandits with Many Arms](https://mlsanity.com/paper/2210.00340/) (2022)

## Abstract (arXiv)

The problem of low-rank matrix completion with heterogeneous and sub-exponential (as opposed to homogeneous and Gaussian) noise is particularly relevant to a number of applications in modern commerce. Examples include panel sales data and data collected from web-commerce systems such as recommendation engines. An important unresolved question for this problem is characterizing the distribution of estimated matrix entries under common low-rank estimators. Such a characterization is essential to any application that requires quantification of uncertainty in these estimates and has heretofore only been available under the assumption of homogenous Gaussian noise. Here we characterize the distribution of estimated matrix entries when the observation noise is heterogeneous sub-exponential and provide, as an application, explicit formulas for this distribution when observed entries are Poisson or Binary distributed.