# Online Nonnegative Matrix Factorization with Outliers

arXiv: https://arxiv.org/abs/1604.02634 · HTML: https://mlsanity.com/paper/1604.02634/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Apr 2016.

## Built on by (most cited first, 7 of 7)

- [GINopic: Topic Modeling with Graph Isomorphism Network](https://mlsanity.com/paper/2404.02115/) (2024)
- [Adversarially-Trained Nonnegative Matrix Factorization](https://mlsanity.com/paper/2104.04757/) (2021)
- [Privacy-preserving Non-negative Matrix Factorization with Outliers](https://mlsanity.com/paper/2211.01451/) (2022)
- [Crime Hot-Spot Modeling via Topic Modeling and Relative Density Estimation](https://mlsanity.com/paper/2202.04176/) (2022)
- [Online Learning Under A Separable Stochastic Approximation Framework](https://mlsanity.com/paper/2305.07484/) (2023)
- [Anomaly Detection in Networked Bandits](https://mlsanity.com/paper/2508.20076/) (2025)
- [Automating Historical Insight Extraction from Large-Scale Newspaper Archives via Neural Topic Modeling](https://mlsanity.com/paper/2512.11635/) (2025)

## Abstract (arXiv)

We propose a unified and systematic framework for performing online nonnegative matrix factorization in the presence of outliers. Our framework is particularly suited to large-scale data. We propose two solvers based on projected gradient descent and the alternating direction method of multipliers. We prove that the sequence of objective values converges almost surely by appealing to the quasi-martingale convergence theorem. We also show the sequence of learned dictionaries converges to the set of stationary points of the expected loss function almost surely. In addition, we extend our basic problem formulation to various settings with different constraints and regularizers. We also adapt the solvers and analyses to each setting. We perform extensive experiments on both synthetic and real datasets. These experiments demonstrate the computational efficiency and efficacy of our algorithms on tasks such as (parts-based) basis learning, image denoising, shadow removal and foreground-background separation.