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Linear spectral unmixing is a popular tool in remotely sensed hyperspectral data interpretation. It aims at estimating the fractional abundances of pure spectral signatures (also called as endmembers) in each mixed pixel collected by an imaging spectrometer. In many situations, the identification of the end-member signatures in the original data set may be challenging due to insufficient spatial resolution, mixtures happening at different scales, and unavailability of completely pure spectral signatures in the scene. However, the unmixing problem can also be approached in semisupervised fashion, i.e., by assuming that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectroradiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In practice, this is a combinatorial problem which calls for efficient linear sparse regression (SR) techniques based on sparsity-inducing regularizers, since the number of endmembers participating in a mixed pixel is usually very small compared with the (ever-growing) dimensionality (and availability) of spectral libraries. Linear SR is an area of very active research, with strong links to compressed sensing, basis pursuit (BP), BP denoising, and matching pursuit. In this paper, we study the linear spectral unmixing problem under the light of recent theoretical results published in those referred to areas. Furthermore, we provide a comparison of several available and new linear SR algorithms, with the ultimate goal of analyzing their potential in solving the spectral unmixing problem by resorting to available spectral libraries. Our experimental results, conducted using both simulated and real hyperspectral data sets collected by the NASA Jet Propulsion Laboratory's Airborne Visible I- - nfrared Imaging Spectrometer and spectral libraries publicly available from the U.S. Geological Survey, indicate the potential of SR techniques in the task of accurately characterizing the mixed pixels using the library spectra. This opens new perspectives for spectral unmixing, since the abundance estimation process no longer depends on the availability of pure spectral signatures in the input data nor on the capacity of a certain endmember extraction algorithm to identify such pure signatures.