Demixing problems in spectroscopic imaging such as hyperspectral imaging and differential optical absorption spectroscopy (DOAS) often require finding sparse nonnegative linear combinations of library functions to match observed data. Due to misalignment and uncertainty issues in data measurement, the known library functions may not represent the data as well as their proper deformations. To improve data adaptivity, we expand the library to one with a group structure and impose a structured sparsity constraint so that the coefficients for each group should be sparse or even 1-sparse. Since the expanded library is often highly coherent, it is difficult to obtain good solutions using convex or greedy methods, such as non-negative least squares (NNLS) or orthogonal matching pursuit. We study efficient penalties related to the Hoyer measure, the ratio of L1 and L2 norms, as sparsity penalties to be added to the objective in NNLS-type models. For solving the resulting non-convex models, we develop a scaled gradient projection algorithm that requires solving a sequence of strongly convex quadratic programs. We discuss its close connections to convex splitting methods and difference of convex programming, and show promising numerical results for DOAS and hyperspectral demixing problems. This is joint work with E. Esser and Y. Lou.
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