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Abstract Due to low spatial resolution, hyperspectral data often consist of mixtures of contributions from multiple materials. This limitation motivates the task of hyperspectral unmixing (HU), which seeks to identify the spectral signatures (endmembers) of the materials present in an observed scene, along with their relative proportions (fractional abundance) in each pixel. A major challenge in HU arises from class variability among materials, which undermines accurate representation using a single spectral signature, as assumed in the conventional linear mixing model. 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$$\ell _1$$ ${\ell }_1$ (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. 

Robust principal component analysis (RPCA) aims to recover a low-rank structure from noisy, partially observed data that is also corrupted by sparse, potentially large-magnitude outliers. Traditional RPCA models rely on convex relaxations, such as nuclear norm and $$ℓ_1$$ norm, to approximate the rank of a matrix and the ℓ0 functional (the number of non-zero elements) of another. In this work, we advocate a nonconvex regularization method, referred to as transformed $$ℓ_1$$ (TL1), to improve both approximations. The rationale is that by varying the internal parameter of TL1, its behavior asymptotically approaches either $$ℓ_0$$ or $$ℓ_1$$ . Since the rank is equal to the number of non-zero singular values and the nuclear norm is defined as their sum, applying TL1 to the singular values can approximate either the rank or the nuclear norm, depending on its internal parameter. We conduct a fine-grained theoretical analysis of statistical convergence rates, measured in the Frobenius norm, for both the low-rank and sparse components under general sampling schemes. These rates are comparable to those of the classical RPCA model based on the nuclear norm and $$ℓ_1$$ norm. Moreover, we establish constant-order upper bounds on the estimated rank of the low-rank component and the cardinality of the sparse component in the regime where TL1 behaves like $$ℓ_0,$$ assuming that the respective matrices are exactly low-rank and exactly sparse. Extensive numerical experiments on synthetic data and real-world applications demonstrate that the proposed approach achieves higher accuracy than the classic convex model, especially under non-uniform sampling schemes. 

Abstract This paper studies a statistical learning model where the model coefficients have a pre-determined non-overlapping group sparsity structure. We consider a combination of a loss function and a regularizer to recover the desired group sparsity patterns, which can embrace many existing works. We analyze directional stationary solutions of the proposed formulation, obtaining a sufficient condition for a directional stationary solution to achieve optimality and establishing a bound of the distance from the solution to a reference point. We develop an efficient algorithm that adopts an alternating direction method of multiplier (ADMM), showing that the iterates converge to a directional stationary solution under certain conditions. In the numerical experiment, we implement the algorithm for generalized linear models with convex and nonconvex group regularizers to evaluate the model performance on various data types, noise levels, and sparsity settings.