Abstract
Optimization methods play a central role in the solution of a wide array of problems encountered in various application fields, such as signal and image processing. Especially when the problems are highly dimensional, proximal methods have shown their efficiency through their capability to deal with composite, possibly nonsmooth objective functions. The cornerstone of these approaches is the proximity operator, which has become a quite popular tool in optimization. In this work, we propose new dual forward-backward formulations for computing the proximity operator of a sum of convex functions involving linear operators. The proposed algorithms are accelerated thanks to the introduction of a block-coordinate strategy combined with a preconditioning technique. Numerical simulations emphasize the good performance of our approach for the problem of jointly deconvoluting and deinterlacing video sequences.
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It is sometimes said that A is full row rank.
References
Abboud, F., Chouzenoux, E., Pesquet, J.C., Chenot, J.H., Laborelli, L.: A hybrid alternating proximal method for blind video restoration. In: 22nd IEEE 22nd European Signal Processing Conference (EUSIPCO 2014), pp. 1811–1815. Lisbon, Portugal (2014)
Bauschke, H.H., Combettes, P.L.: A Dykstra-like algorithm for two monotone operators. Pac. J. Optim. 4(3), 383–391 (2008)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Bolte, J., Nguyen, T.P., Peypouquet, J., Suter, B.: From error bounds to the complexity of first-order descent methods for convex functions. Tech. rep. (2015). http://www.optimization-online.org/DB_HTML/2015/10/5176.html
Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1–2), 459–494 (2014)
Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 8(1), 1–122 (2011)
Chambolle, A., Pock, T.: A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions. SMAI J. Comput. Math. 1, 29–54 (2015)
Chouzenoux, E., Pesquet, J.C., Repetti, A.: Variable metric forward-backward algorithm for minimizing the sum of a differentiable function and a convex function. J. Optim. Theory App. 162(1), 107–132 (2014)
Chouzenoux, E., Pesquet, J.C., Repetti, A.: A block coordinate variable metric forward-backward algorithm. J. Global Optim. 1–29 (2016)
Combettes, P.L., Dũng, D., Vũ, B.C.: Proximity for sums of composite functions. J. Math. Anal. Appl. 380(2), 680–688 (2011)
Combettes, P.L., Dung, D., Vũ, B.C.: Dualization of signal recovery problems. Set-Valued Var. Anal. 18(3), 373–404 (2010)
Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer-Verlag, New York (2010)
Combettes, P.L., Pesquet, J.C.: Stochastic quasi-Fejér block-coordinate fixed point iterations with random sweeping. SIAM J. Optim. 25(2), 1221–1248 (2015)
Combettes, P.L., Vũ, B.C.: Variable metric forward-backward splitting with applications to monotone inclusions in duality. Optimization 63(9), 1289–1318 (2014)
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)
Condat, L.: Semi-local total variation for regularization of inverse problems. In: 22nd IEEE European Signal Processing Conference (EUSIPCO 2014), pp. 1806–1810. Lisbon, Portugal (2014)
Csiba, D., Qu, Z., Richtárik, P.: Stochastic dual coordinate ascent with adaptive probabilities. In: 32nd International Conference on Machine Learning (ICML 2015), pp. 674–683. Lille, France (2015)
Fu, X., He, B., Wang, X., Yuan, X.: Block wise alternating direction method of multipliers with gaussian back substitution for multiple block convex programming (2014). http://www.optimization-online.org/DB_FILE/2014/09/4544.pdf
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Jaggi, M., Smith, V., Takác, M., Terhorst, J., Krishnan, S., Hofmann, T., Jordan, M.I.: Communication-efficient distributed dual coordinate ascent. In: Advances in and Neural Information Processing Systems (NIPS 2014), pp. 3068–3076. Montréal, Canada (2014)
Keller, S.H.: Video upscaling using variational methods. Ph.D. thesis, The Image Group, Department of Computer Science Faculty of Science, University of Copenhagen (2007)
Keller, S.H., Lauze, F., Nielsen, M.: A total variation motion adaptive deinterlacing scheme. In: 5th International Conference Scale-Space 2005, pp. 408–418. Hofgeismar, Germany (2005)
Krishnan, D., Tay, T., Fergus, R.: Blind deconvolution using a normalized sparsity measure. In: IEEE Conference Computer Vision Pattern Recognition (CVPR 2011), pp. 233–240. Colorado Springs, CO (2011)
Liu, C., Freeman, W.T., Adelson, E.H., Weiss, Y.: Human-assisted motion annotation. In: IEEE Conference Computer Vision Pattern Recognition (CVPR 2008), pp. 1–8. Anchorage, Alaska (2008)
Lojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. Editions du centre National de la Recherche Scientifique pp. 87–89 (1963)
Lorenz, D.A., Wenger, S., Schöpfer, F., Magnor, M.A.: A sparse Kaczmarz solver and a linearized Bregman method for online compressed sensing. In: IEEE International Conference on Image Processing (ICIP 2014), pp. 1347–1351. Paris, France (2014)
Mallat, S.: Super resolution bandlet upconversion for HD tv. Tech. rep. (2006). http://www.di.ens.fr/~mallat/papiers/whitepaper.pdf
Pesquet, J.C., Repetti, A.: A class of randomized primal-dual algorithms for distributed optimization. J. Nonlinear Convex Anal. 16(12), 2453–2490 (2015)
Qu, Z., Richtárik, P., Zhang, T.: Randomized dual coordinate ascent with arbitrary sampling. In: Advances in Neural Information Processing System (NIPS 2015), pp. 865–873. Montréal, Canada (2015)
Richtárik, P., Takác, M.: Efficient serial and parallel coordinate descent methods for huge-scale truss topology design. In: International Conference on Operations Research (OR 2011), pp. 27–32 (2011)
Saha, A., Tewari, A.: On the finite time convergence of cyclic coordinate descent methods. SIAM J. Optim. 23(1), 576–601 (2013)
Seshadrinathan, k, Bovik, A.C.: Motion tuned spatio-temporal quality assessment of natural videos. IEEE Trans. Image Process. 19(2), 335–350 (2010)
Shalev-Shwartz, S., Zhang, T.: Stochastic dual coordinate ascent methods for regularized loss minimization. J. Mach. Learn. Res. 14(1), 567–599 (2013)
Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)
Xu, Z., Gan, Z., Zhu, X.: Compressed video super-resolution reconstruction based on regularized algorithm. In: 6th IEEE International Conference Signal Processing (ICSP 2006). Beijing, China (2006)
Zhang, X., Xiong, R., Ma, S., Gao, W.: A robust video super-resolution algorithm. In: 28th IEEE Picture Coding Symposium (PCS 2010), pp. 574–577. Nagoya, Japan (2010)
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Abboud, F., Chouzenoux, E., Pesquet, JC. et al. Dual Block-Coordinate Forward-Backward Algorithm with Application to Deconvolution and Deinterlacing of Video Sequences. J Math Imaging Vis 59, 415–431 (2017). https://doi.org/10.1007/s10851-016-0696-y
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DOI: https://doi.org/10.1007/s10851-016-0696-y