Biobjective Nonnegative Matrix Factorization: Linear Versus Kernel-Based Models

Abstract

Abstract-Nonnegative matrix factorization (NMF) is a powerful class of feature extraction techniques that has been successfully applied in many fields, namely in signal and image processing. Current NMF techniques have been limited to a single-objective problem in either its linear or nonlinear kernelbased formulation. In this paper, we propose to revisit the NMF as a multi-objective problem, in particular a bi-objective one, where the objective functions defined in both input and feature spaces are taken into account. By taking the advantage of the sum-weighted method from the literature of multi-objective optimization, the proposed bi-objective NMF determines a set of nondominated, Pareto optimal, solutions instead of a single optimal decomposition. Moreover, the corresponding Pareto front is studied and approximated. Experimental results on unmixing real hyperspectral images confirm the efficiency of the proposed bi-objective NMF compared with the state-of-the-art methods.

Unsupervised Nonlinear Hyperspectral Unmixing Based on Bilinear Mixture Models via Geometric Projection and Constrained Nonnegative Matrix Factorization

“…Accordingly, four supervised algorithms, including the famous linear algorithm FCLS [6], GBM-based semiNMF [38], PPNM-based GDA [32], and MLM [37] are compared for the abundance estimation by using true endmembers or the endmembers extracted by VCA [4]. Besides, four state-of-the-art unsupervised nonlinear spectral unmixing algorithms, including the Fan-NMF [30], distance-based nonlinear simplex projection unmixing (DNSPU) [46], RNMF [54], and bi-objective kernel NMF (Bio-KNMF) [57] have also been considered. Most of these algorithms have been briefly described in the introduction.…”

“…Some existing algorithms such as the multiple update rule [13] and projected gradient (PG) method [48,57,68] can solve the optimization problem in (12). In this paper, the PG method is adopted to update endmembers A and abundances S alternately in the framework of constrained NMF.…”

“…α t and β t are the updating steps which can be determined using the Armijo rule [48,57,68]. In simple terms, α t and β t decrease to give a better convergence when the relative error condition between two contiguous iterations in [68] is satisfied, or they will increase to accelerate the search.…”

“…Update endmembers and abundances in the constrained NMF framework (2a) Update S t+1 with Equation (16) [48,57,68]. (2b) Update A t+1 with Equation (17) [48,57,68].…”

“…In this ideal situation, BCNMF shows better or competitive Fan-NMF [48] GBM-semiNMF [38] PPNMGDA [32] MLM [37] DNSPU [46] RNMF [54] Bio-KNMF [57] BCNMF (FM) BCNMF (GBM) BCNMF (PPNM) Figure 19. The differences in trees' abundance maps between the fully constrained least squares (FCLS) and nonlinear spectral unmixing algorithms.…”