A General Framework for Group Sparsity in Hyperspectral Unmixing Using Endmember Bundles

Due to low spatial resolution, hyperspectral data often consists of mixtures of contributions from multiple materials. This limitation motivates the task of hyperspectral unmixing (HU), a fundamental problem in hyperspectral imaging. HU aims to identify the spectral signatures (\textit{endmembers}) of the materials present in an observed scene, along with their relative proportions (\textit{fractional abundance}) in each pixel. A major challenge lies in the class variability in materials, which hinders accurate representation by a single spectral signature, as assumed in the conventional linear mixing model. Moreover, To address this issue, we propose using group sparsity after representing each material with a set of spectral signatures, known as endmember bundles, where each group corresponds to a specific material. In particular, we develop a bundle-based framework that can enforce either inter-group sparsity or sparsity within and across groups (SWAG) on the abundance coefficients. Furthermore, our framework offers the flexibility to incorporate a variety of sparsity-promoting penalties, among which the transformed $\ell_1$ (TL1) penalty is a novel regularization in the HU literature. Extensive experiments conducted on both synthetic and real hyperspectral data demonstrate the effectiveness and superiority of the proposed approaches.

MALADY: Multiclass Active Learning with Auction Dynamics on Graphs

Active learning enhances the performance of machine learning methods, particularly in semi-supervised cases, by judiciously selecting a limited number of unlabeled data points for labeling, with the goal of improving the performance of an underlying classifier. In this work, we introduce the Multiclass Active Learning with Auction Dynamics on Graphs (MALADY) framework which leverages the auction dynamics algorithm on similarity graphs for efficient active learning. In particular, we generalize the auction dynamics algorithm on similarity graphs for semi-supervised learning in [24] to incorporate a more general optimization functional. Moreover, we introduce a novel active learning acquisition function that uses the dual variable of the auction algorithm to measure the uncertainty in the classifier to prioritize queries near the decision boundaries between different classes. Lastly, using experiments on classification tasks, we evaluate the performance of our proposed method and show that it exceeds that of comparison algorithms.

Persistent Laplacian-enhanced Algorithm for Scarcely Labeled Data Classification

The success of many machine learning (ML) methods depends crucially on having large amounts of labeled data. However, obtaining enough labeled data can be expensive, time-consuming, and subject to ethical constraints for many applications. One approach that has shown tremendous value in addressing this challenge is semi-supervised learning (SSL); this technique utilizes both labeled and unlabeled data during training, often with much less labeled data than unlabeled data, which is often relatively easy and inexpensive to obtain. In fact, SSL methods are particularly useful in applications where the cost of labeling data is especially expensive, such as medical analysis, natural language processing (NLP), or speech recognition. A subset of SSL methods that have achieved great success in various domains involves algorithms that integrate graph-based techniques. These procedures are popular due to the vast amount of information provided by the graphical framework and the versatility of their applications. In this work, we propose an algebraic topology-based semi-supervised method called persistent Laplacian-enhanced graph MBO (PL-MBO) by integrating persistent spectral graph theory with the classical Merriman-Bence- Osher (MBO) scheme. Specifically, we use a filtration procedure to generate a sequence of chain complexes and associated families of simplicial complexes, from which we construct a family of persistent Laplacians. Overall, it is a very efficient procedure that requires much less labeled data to perform well compared to many ML techniques, and it can be adapted for both small and large datasets. We evaluate the performance of the proposed method on data classification, and the results indicate that the proposed technique outperforms other existing semi-supervised algorithms.