Quaternion Non-Negative Matrix Factorization: Definition, Uniqueness, and Algorithm

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