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A conjugate subgradient algorithm with adaptive preconditioning for the least absolute shrinkage and selection operator minimization
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 4 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper describes a new efficient conjugate subgradient algorithm which minimizes a convex function containing a least squares fidelity term and an absolute value regularization term. This method is successfully applied to the inversion of ill-conditioned linear problems, in particular for computed tomography with the dictionary learning method. A comparison with other state-of-art methods shows a significant reduction of the number of iterations, which makes this algorithm appealing for practical use. 
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     title = {A conjugate subgradient algorithm with adaptive preconditioning for the least absolute shrinkage and selection operator minimization},
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A. Mirone; P. Paleo. A conjugate subgradient algorithm with adaptive preconditioning for the least absolute shrinkage and selection operator minimization. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 4. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_4_a12/

[1] F. Bach, R. Jenatton, J. Mairal, G. Obozinski, “Optimization with sparsity-inducing penalties”, Foundations Trends Machine Learning, 4:1 (2011), 1–106 | DOI | MR | Zbl

[2] A. N. Tikhonov, “On the stability of inverse problems”, Dokl. Akad. Nauk SSSR, 39:5 (1943), 195–198 | MR

[3] A. Chambolle, T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging”, J. Math. Imaging Vision, 40:1 (2011), 120–145 | DOI | MR | Zbl

[4] E. Y. Sidky, J. H. Jorgensen, X. Pan, “Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm”, Phys. Med. Biol., 57:10 (2012), 3065–3091 | DOI

[5] P. L. Combettes, J.-C. Pesquet, Proximal Splitting Methods in Signal Processing, 2009, arXiv: 0912.3522 | MR

[6] R. Tibshirani, “Regression shrinkage and selection via the lasso”, J. R. Stat. Soc. Ser. B (Methodological), 58:1 (1996), 267–288 | MR | Zbl

[7] L. I. Rudin, S. Osher, E. Fatemi, “Nonlinear total variation based noise removal algorithms”, Phys. D: Nonlinear Phenom., 60:1–4 (1992), 259–268 | DOI | MR | Zbl

[8] I. W. Selesnick, M. A. T. Figueiredo, “Signal restoration with overcomplete wavelet transforms: Comparison of analysis and synthesis priors”, Wavelets XIII, Proc. SPIE, 7446, 2009, 74460D | DOI | MR

[9] A. Beck, M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems”, IEEE Trans. Image Process., 18 (2009), 2419–2434 | DOI | MR

[10] Y. Nesterov, “A method of solving a convex programming problem with convergence rate $O(1/sqr(k))$”, Sov. Math. Dokl., 27 (1983), 372–376 | MR | Zbl

[11] S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers”, Found. Trends Machine Learn., 3:1 (2011), 1–122 | DOI

[12] S. Boyd, L. Xiao, A. Mutapcic, Subgradient methods, Notes for EE, No 3920, Stanford Univ., 2003

[13] S. Bubeck, Theory of convex optimization for machine learning, 2014, arXiv: 1405.4980

[14] A. Mirone, P. Paleo, Python script: Csg.py, https://github.com/pierrepaleo/csg

[15] A. Mirone, E. Brun, P. Coan, “A dictionary learning approach with overlap for the low dose computed tomography reconstruction and its vectorial application to differential phase tomography”, PLoS ONE, 9:12 (2014), e114325 | DOI

[16] E. J. Candes, J. K. Romberg, T. Tao, “Stable signal recovery from incomplete and inaccurate measurements”, Commun. Pure Appl. Math., 59:8 (2006), 1207–1223 | DOI | MR | Zbl

[17] P. Paleo, A. Mirone, Ring artifacts correction in compressed sensing tomographic reconstruction, 2015, arXiv: 1502.01480

[18] A. Mirone, E. Brun, E. Gouillart, P. Tafforeau, J. Kieffer, “The PyHST2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities”, Nuclear Instrum. Methods Phys. Res. Sect B: Beam Interactions Materials Atoms, 324 (2014), 41–48 | DOI