# Quaternion Non-negative Matrix Factorization: definition, uniqueness and algorithm

arXiv: https://arxiv.org/abs/1903.10593 · HTML: https://mlsanity.com/paper/1903.10593/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Mar 2019.

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

- [Quasi Non-Negative Quaternion Matrix Factorization with Application to Color Face Recognition](https://mlsanity.com/paper/2211.16739/) (2022)
- [Separable Quaternion Matrix Factorization for Polarization Images](https://mlsanity.com/paper/2207.14039/) (2022)
- [Convex Quaternion Optimization for Signal Processing: Theory and Applications](https://mlsanity.com/paper/2305.06879/) (2023)
- [Non-Negative Reduced Biquaternion Matrix Factorization with Applications in Color Face Recognition](https://mlsanity.com/paper/2408.05582/) (2024)

## Abstract (arXiv)

This article introduces quaternion non-negative matrix factorization (QNMF), which generalizes the usual non-negative matrix factorization (NMF) to the case of polarized signals. Polarization information is represented by Stokes parameters, a set of 4 energetic parameters widely used in polarimetric imaging. QNMF relies on two key ingredients: (i) the algebraic representation of Stokes parameters thanks to quaternions and (ii) the exploitation of physical constraints on Stokes parameters. These constraints generalize non-negativity to the case of polarized signals, encoding positive semi-definiteness of the covariance matrix associated which each source. Uniqueness conditions for QNMF are presented. Remarkably, they encompass known sufficient uniqueness conditions from NMF. Meanwhile, QNMF further relaxes NMF uniqueness conditions requiring sources to exhibit a certain zero-pattern, by leveraging the complete polarization information. We introduce a simple yet efficient algorithm called quaternion alternating least squares (QALS) to solve the QNMF problem in practice. Closed-form quaternion updates are obtained using the recently introduced generalized HR calculus. Numerical experiments on synthetic data demonstrate the relevance of the approach. QNMF defines a promising generic low-rank approximation tool to handle polarization, notably for blind source separation problems arising in imaging applications.