Kernel Nonnegative Matrix Factorization Without the Curse of the Pre-image - Application to Unmixing Hyperspectral Images
This addresses a challenge in nonlinear NMF for applications like hyperspectral image analysis, offering a practical solution to the curse of the pre-image, though it appears incremental as it builds on existing kernel and NMF frameworks.
The paper tackles the pre-image problem in kernel nonnegative matrix factorization (NMF) by proposing a novel kernel-based model that estimates factorization matrices directly in the input space, and demonstrates its effectiveness in unmixing hyperspectral images with competitive results against state-of-the-art techniques.
The nonnegative matrix factorization (NMF) is widely used in signal and image processing, including bio-informatics, blind source separation and hyperspectral image analysis in remote sensing. A great challenge arises when dealing with a nonlinear formulation of the NMF. Within the framework of kernel machines, the models suggested in the literature do not allow the representation of the factorization matrices, which is a fallout of the curse of the pre-image. In this paper, we propose a novel kernel-based model for the NMF that does not suffer from the pre-image problem, by investigating the estimation of the factorization matrices directly in the input space. For different kernel functions, we describe two schemes for iterative algorithms: an additive update rule based on a gradient descent scheme and a multiplicative update rule in the same spirit as in the Lee and Seung algorithm. Within the proposed framework, we develop several extensions to incorporate constraints, including sparseness, smoothness, and spatial regularization with a total-variation-like penalty. The effectiveness of the proposed method is demonstrated with the problem of unmixing hyperspectral images, using well-known real images and results with state-of-the-art techniques.