# On Mixed Memberships and Symmetric Nonnegative Matrix Factorizations

arXiv: https://arxiv.org/abs/1607.00084 · HTML: https://mlsanity.com/paper/1607.00084/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2016.

## Built on by (most cited first, 14 of 14)

- [Memory-Efficient Convex Optimization for Self-Dictionary Separable Nonnegative Matrix Factorization: A Frank-Wolfe Approach](https://mlsanity.com/paper/2109.11135/) (2021)
- [Deep Clustering with Incomplete Noisy Pairwise Annotations: A Geometric Regularization Approach](https://mlsanity.com/paper/2305.19391/) (2023)
- [Bipartite mixed membership distribution-free model. A novel model for community detection in overlapping bipartite weighted networks](https://mlsanity.com/paper/2211.00912/) (2022)
- [A useful criterion on studying consistent estimation in community detection](https://mlsanity.com/paper/2109.14950/) (2021)
- [Directed mixed membership stochastic blockmodel](https://mlsanity.com/paper/2101.02307/) (2021)
- [Graph Community Detection from Coarse Measurements: Recovery Conditions for the Coarsened Weighted Stochastic Block Model](https://mlsanity.com/paper/2102.13135/) (2021)
- [Directed degree corrected mixed membership model and estimating community memberships in directed networks](https://mlsanity.com/paper/2109.07826/) (2021)
- [Mixed membership distribution-free model](https://mlsanity.com/paper/2112.04389/) (2021)
- [Mixed membership estimation for categorical data with weighted responses](https://mlsanity.com/paper/2310.10989/) (2023)
- [Matrix Factorization for Inferring Associations and Missing Links](https://mlsanity.com/paper/2503.04680/) (2025)
- [Fuzzy Clustering with Similarity Queries](https://mlsanity.com/paper/2106.02212/) (2021)
- [Robust Correlation Clustering with Asymmetric Noise](https://mlsanity.com/paper/2110.08385/) (2021)
- [Accelerated structured matrix factorization](https://mlsanity.com/paper/2212.06504/) (2022)
- [A Hybrid Membership Latent Distance Model for Unsigned and Signed Integer Weighted Networks](https://mlsanity.com/paper/2308.15293/) (2023)

## Abstract (arXiv)

The problem of finding overlapping communities in networks has gained much attention recently. Optimization-based approaches use non-negative matrix factorization (NMF) or variants, but the global optimum cannot be provably attained in general. Model-based approaches, such as the popular mixed-membership stochastic blockmodel or MMSB (Airoldi et al., 2008), use parameters for each node to specify the overlapping communities, but standard inference techniques cannot guarantee consistency. We link the two approaches, by (a) establishing sufficient conditions for the symmetric NMF optimization to have a unique solution under MMSB, and (b) proposing a computationally efficient algorithm called GeoNMF that is provably optimal and hence consistent for a broad parameter regime. We demonstrate its accuracy on both simulated and real-world datasets.