Bi-Objective Nonnegative Matrix Factorization: Linear Versus Kernel-Based Models

TL;DR

This paper introduces a bi-objective nonnegative matrix factorization approach that considers both linear and kernel-based models, producing a set of Pareto optimal solutions and demonstrating improved performance on hyperspectral image unmixing.

Contribution

It formulates NMF as a bi-objective problem, enabling the generation of Pareto optimal solutions and analyzing the Pareto front, which is a novel approach in NMF research.

Findings

Bi-objective NMF outperforms state-of-the-art methods in hyperspectral unmixing.

The Pareto front provides a spectrum of optimal solutions for different trade-offs.

Experimental results validate the effectiveness of the proposed approach.

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 kernel-based 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.