Implementation of an optimal first-order method for strongly convex total variation regularization |
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Authors: | T. L. Jensen J. H. J?rgensen P. C. Hansen S. H. Jensen |
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Affiliation: | 1. Department of Electronic Systems, Aalborg University, Niels Jernesvej 12, 9220, Aalborg ?, Denmark 2. Department of Informatics and Mathematical Modelling, Technical University of Denmark, Building 321, 2800, Lyngby, Denmark
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Abstract: | We present a practical implementation of an optimal first-order method, due to Nesterov, for large-scale total variation regularization in tomographic reconstruction, image deblurring, etc. The algorithm applies to μ-strongly convex objective functions with L-Lipschitz continuous gradient. In the framework of Nesterov both μ and L are assumed known—an assumption that is seldom satisfied in practice. We propose to incorporate mechanisms to estimate locally sufficient μ and L during the iterations. The mechanisms also allow for the application to non-strongly convex functions. We discuss the convergence rate and iteration complexity of several first-order methods, including the proposed algorithm, and we use a 3D tomography problem to compare the performance of these methods. In numerical simulations we demonstrate the advantage in terms of faster convergence when estimating the strong convexity parameter μ for solving ill-conditioned problems to high accuracy, in comparison with an optimal method for non-strongly convex problems and a first-order method with Barzilai-Borwein step size selection. |
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