Journal
SIAM JOURNAL ON OPTIMIZATION
Volume 18, Issue 4, Pages 1351-1376Publisher
SIAM PUBLICATIONS
DOI: 10.1137/060669498
Keywords
convex programming; deconvolution; denoising; forward-backward splitting algorithm; Hilbert space; orthonormal basis; proximal algorithm; proximal thresholding; proximity operator; signal recovery; soft thresholding; strong convergence
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The notion of soft thresholding plays a central role in problems from various areas of applied mathematics, in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties, providing, in particular, several characterizations of such thresholders. We then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems, and we establish the strong convergence of a proximal thresholding algorithm to solve it. Numerical applications to signal recovery are demonstrated.
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