# Fixed points of the EM algorithm and nonnegative rank boundaries

arXiv: https://arxiv.org/abs/1312.5634 · HTML: https://mlsanity.com/paper/1312.5634/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Dec 2013.

## Built on by (most cited first, 2 of 2)

- [The Nondecreasing Rank](https://mlsanity.com/paper/2509.00265/) (2025)
- [Robustness of Minimum-Volume Nonnegative Matrix Factorization under an Expanded Sufficiently Scattered Condition](https://mlsanity.com/paper/2511.04291/) (2025)

## Abstract (arXiv)

Mixtures of r independent distributions for two discrete random variables can be represented by matrices of nonnegative rank r. Likelihood inference for the model of such joint distributions leads to problems in real algebraic geometry that are addressed here for the first time. We characterize the set of fixed points of the Expectation-Maximization algorithm, and we study the boundary of the space of matrices with nonnegative rank at most 3. Both of these sets correspond to algebraic varieties with many irreducible components.