Hyperspectral imaging technology has a wide range of applications, including forest management, mineral resource exploration, and Earth surface monitoring. Endmember extraction of hyperspectral images is a key step in leveraging this technology for applications. It aims to identifying the spectral signatures of materials, i.e., the major components in the observed scenes. Theoretically speaking, Hottopixx methods should be effective on problems involving extracting endmembers from hyperspectral images. Yet, these methods are challenging to perform in practice, due to high computational costs. They require us to solve LP problems, called Hottopixx models, whose size grows quadratically with the number of pixels in the image. It is thus still unclear as to whether they are actually effective or not. This study clarifies this situation. We propose an efficient and effective implementation of Hottopixx. Our implementation follows the framework of column generation, which is known as a classical but powerful means of solving large-scale LPs. We show in experiments that our implementation is applicable to the endmember extraction from real hyperspectral images and can provide estimations of endmember signatures with higher accuracy than the existing methods can.

Deep Nonnegative Matrix Factorization (deep NMF) has recently emerged as a valuable technique for extracting multiple layers of features across different scales. However, all existing deep NMF models and algorithms have primarily centered their evaluation on the least squares error, which may not be the most appropriate metric for assessing the quality of approximations on diverse datasets. For instance, when dealing with data types such as audio signals and documents, it is widely acknowledged that $\beta$-divergences offer a more suitable alternative. In this paper, we develop new models and algorithms for deep NMF using $\beta$-divergences. Subsequently, we apply these techniques to the extraction of facial features, the identification of topics within document collections, and the identification of materials within hyperspectral images.

Constrained low-rank matrix approximations have been known for decades as powerful linear dimensionality reduction techniques to be able to extract the information contained in large data sets in a relevant way. However, such low-rank approaches are unable to mine complex, interleaved features that underlie hierarchical semantics. Recently, deep matrix factorization (deep MF) was introduced to deal with the extraction of several layers of features and has been shown to reach outstanding performances on unsupervised tasks. Deep MF was motivated by the success of deep learning, as it is conceptually close to some neural networks paradigms. In this paper, we present the main models, algorithms, and applications of deep MF through a comprehensive literature review. We also discuss theoretical questions and perspectives of research.

Hyperspectral Unmixing (HU) has received increasing attention in the past decades due to its ability of unveiling information latent in hyperspectral data. Unfortunately, most existing methods fail to take advantage of the spatial information in data. To overcome this limitation, we propose a Structured Sparse regularized Nonnegative Matrix Factorization (SS-NMF) method from the following two aspects. First, we incorporate a graph Laplacian to encode the manifold structures embedded in the hyperspectral data space. In this way, the highly similar neighboring pixels can be grouped together. Second, the lasso penalty is employed in SS-NMF for the fact that pixels in the same manifold structure are sparsely mixed by a common set of relevant bases. These two factors act as a new structured sparse constraint. With this constraint, our method can learn a compact space, where highly similar pixels are grouped to share correlated sparse representations. Experiments on real hyperspectral data sets with different noise levels demonstrate that our method outperforms the state-of-the-art methods significantly.