Toward Convergence: A Gradient-Based Multiobjective Method With Greedy Hash for Hyperspectral Unmixing

Multiobjective optimization aims at addressing the conflicting objectives, which has been introduced to improve the performance of sparse hyperspectral unmixing. Recently proposed multiobjective unmixing methods usually employ evolutionary algorithms (EAs) to improve the unmixing accuracy. However, EAs may suffer the challenge of convergence, in which case the reasonability of the solutions is hard to guarantee. To solve the problem of convergence, in this article, we present a new gradient-based multiobjective unmixing method that explores the optimization direction in a theoretically reliable manner. Furthermore, considering the mathematical model of hyperspectral sparse unmixing where the sparsity error objective of selected endmembers is discrete, we develop a greedy hash-based coding approach that is able to well describe the discrete constraints imposed on endmembers. The major components of the proposed method are a search approach and an update approach. In the search approach, we construct the Pareto descent direction via a gradient-based strategy, which contributes to converging to an optimal continuous solution by searching along this direction. In the update approach, we update discrete binary endmember via hash coding under the guidance of the greedy principle, which allows our method to handle the problem of discrete objectives. The major contribution of the proposed method is designing a new framework that can get the optimal discrete endmembers in a convergent way. Moreover, we provide the theoretical analysis and proof for convergence. Synthetic and real-world experiments have indicated the advantages of our algorithm compared with evolutionary multiobjective unmixing methods.