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

Abstract

This article introduces quaternion non-negative matrix factorization (QNMF), which generalizes the usual nonnegative 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.

Quaternion Non-negative Matrix Factorization: a new tool for spectropolarimetric imaging

“…Activation maps are chosen such that the pure pixel condition is satisfied. Such conditions guarantee that the QNMF X 0 = W 0 H 0 satisfies the necessary uniqueness condition given in [6,Prop. 3].…”

“…Importantly, these two matrix factorization problems are not independent, for two reasons: (i) the activation matrix H is a common factor and (ii) the very nature of the polarization constraint (S) links the real and imaginary parts of the source matrix W. In full generality, QNMF greatly improves NMF model identifiablity by taking advantage of supporting polarization informationfundamental properties that can be measured in most optical imaging setups. From a theoretical perspective, we derive in [6] sufficient conditions (for P = 2 sources) and necessary uniqueness conditions (for P ≥ 2 sources). Compared to their NMF counterparts [11]- [14], these conditions appear far less restrictive -in particular, one may recover QNMF uniqueness even when sources never vanish, a case where NMF is known to be not unique.…”

“…Π R+ (H pn ) = max(0, H pn ). The projection of elements of W onto (S) is performed by projecting the associated 2-by-2 Hermitian matrix onto the subspace spanned by its nonnegative eigenvalues, see [6] for details.…”

“…Detailed derivations of explicit updates for unconstrained least-squares problems are provided in [6]. It requires special care due to the quaternion nature of X and W. For H, it involves a cautious handling of quaternion non-commutativity.…”

“…Focusing on practical purposes, we give a simple yet numerically efficient algorithm that solves the QNMF problem using a quaterniondomain alternated least square optimization strategy. A first theoretical analysis of QNMF can be found in [6], where it is shown that QNMF greatly improves NMF in terms of model identifiability, by having less strict uniqueness conditions on the sources. Numerical experiments on synthetic data demonstrate the potential of QNMF as a generic low-rank approximation tool for polarized signals.…”