A Convex Geometry-Based Blind Source Separation Method for Separating Nonnegative Sources

Nonnegative Blind Source Separation for Ill-Conditioned Mixtures via John Ellipsoid

Nonnegative blind source separation (nBSS) is often a challenging inverse problem, namely, when the mixing system is ill-conditioned. In this work, we focus on an important nBSS instance, known as hyperspectral unmixing (HU) in remote sensing. HU is a matrix factorization problem aimed at factoring the so-called endmember matrix, holding the material hyperspectral signatures, and the abundance matrix, holding the material fractions at each image pixel. The hyperspectral signatures are usually highly correlated, leading to a fast decay of the singular values (and, hence, high condition number) of the endmember matrix, so HU often introduces an ill-conditioned nBSS scenario. We introduce a new theoretical framework to attack such tough scenarios via the John ellipsoid (JE) in functional analysis. The idea is to identify the maximum volume ellipsoid inscribed in the data convex hull, followed by affinely mapping such ellipsoid into a Euclidean ball. By applying the same affine mapping to the data mixtures, we prove that the endmember matrix associated with the mapped data has condition number 1, the lowest possible, and that these (preconditioned) endmembers form a regular simplex. Exploiting this regular structure, we design a novel nBSS criterion with a provable identifiability guarantee and devise an algorithm to realize the criterion. Moreover, for the first time, the optimization problem for computing JE is exactly solved for a large-scale instance; our solver employs a split augmented Lagrangian shrinkage algorithm with all proximal operators solved by closed-form solutions. The competitiveness of the proposed method is illustrated by numerical simulations and real data experiments.

Detection of Sources in Non-Negative Blind Source Separation by Minimum Description Length Criterion

While non-negative blind source separation (nBSS) has found many successful applications in science and engineering, model order selection, determining the number of sources, remains a critical yet unresolved problem. Various model order selection methods have been proposed and applied to real-world data sets but with limited success, with both order over- and under-estimation reported. By studying existing schemes, we have found that the unsatisfactory results are mainly due to invalid assumptions, model oversimplification, subjective thresholding, and/or to assumptions made solely for mathematical convenience. Building on our earlier work that reformulated model order selection for nBSS with more realistic assumptions and models, we report a newly and formally revised model order selection criterion rooted in the minimum description length (MDL) principle. Adopting widely invoked assumptions for achieving a unique nBSS solution, we consider the mixing matrix as consisting of deterministic unknowns, with the source signals following a multivariate Dirichlet distribution. We derive a computationally efficient, stochastic algorithm to obtain approximate maximum-likelihood estimates of model parameters and apply Monte Carlo integration to determine the description length. Our modeling and estimation strategy exploits the characteristic geometry of the data simplex in nBSS. We validate our nBSS-MDL criterion through extensive simulation studies and on four real-world data sets, demonstrating its strong performance and general applicability to nBSS. The proposed nBSS-MDL criterion consistently detects the true number of sources, in all of our case studies.

Local Smoothness Constrained Nonnegative Matrix Factorization with Nonlinear Convergence Rate for Spectral Decomposition

With the development of the detection technology using multispectra sensors, spectral decomposition (SD) attracts more and more attention in the biomedical signal processing and image processing. In this paper, a local smoothness constrained nonnegative matrix factorization (NMF) with nonlinear convergence rate (NMF-NCR) is proposed to solve SD problem and our contributions are as follows. First, it proves that the gradients of the cost function with respect to each variable matrix are Lipschitz continuous. Then, a proximal function is constructed for optimizing the cost function. As a result, our method can achieve an NCR much faster than the traditional methods. Simulations show the advantage in solving SD of our algorithm over the compared methods.