Linear hyperspectral unmixing (LHU) aims to decompose mixed pixels into endmembers and abundance vectors, and it has important applications in terrain classification, mineral recognition and quantification, agricultural monitoring, and military surveillance. Because the data-fidelity term couples the endmember and abundance variables, the original formulation is nonconvex. After a principal components analysis (PCA)-based dimensionality reduction and the inverse transformation $ Q = M^{-1} $, we obtain a smooth reduced model on the nonsingular domain. We first apply the adaptive moments method (ADAM) to this transformed LHU model and then develop a stochastic adaptive moments method (SADAM) based on mini-batch gradients to reduce the computational cost for large data sets. For SADAM, we provide a conditional convergence analysis to stationary points under smoothness, boundedness, and moment-tracking assumptions. Numerical experiments on four real hyperspectral data sets show that SADAM can reduce computational time while maintaining competitive unmixing accuracy.
Citation: Zhewei Zhang, Fangfang Xu. Adaptive moments method for linear hyperspectral unmixing[J]. AIMS Mathematics, 2026, 11(10): 32839-32859. doi: 10.3934/math.20261288
Linear hyperspectral unmixing (LHU) aims to decompose mixed pixels into endmembers and abundance vectors, and it has important applications in terrain classification, mineral recognition and quantification, agricultural monitoring, and military surveillance. Because the data-fidelity term couples the endmember and abundance variables, the original formulation is nonconvex. After a principal components analysis (PCA)-based dimensionality reduction and the inverse transformation $ Q = M^{-1} $, we obtain a smooth reduced model on the nonsingular domain. We first apply the adaptive moments method (ADAM) to this transformed LHU model and then develop a stochastic adaptive moments method (SADAM) based on mini-batch gradients to reduce the computational cost for large data sets. For SADAM, we provide a conditional convergence analysis to stationary points under smoothness, boundedness, and moment-tracking assumptions. Numerical experiments on four real hyperspectral data sets show that SADAM can reduce computational time while maintaining competitive unmixing accuracy.
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Zhewei Zhang, Fangfang Xu. Adaptive moments method for linear hyperspectral unmixing[J]. AIMS Mathematics, 2026, 11(10): 32839-32859. doi: 10.3934/math.20261288
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