, 1) holds because x is k-sparse and b |k is the best k-sparse approximation of b |R since supp(b |k ) = supp(x t+1 ) ? R, p.1

B. Adcock and A. C. Hansen, Generalized sampling and infinite-dimensional compressed sensing, Foundations of Computational Mathematics, vol.16, issue.5, pp.1263-1323, 2016.

S. Bahmani, B. Raj, and P. Boufounos, Greedy sparsity-constrained optimization, J. of Machine Learning Research, vol.14, issue.3, pp.807-841, 2013.

H. Bauschke, P. Combettes, and S. Reich, The asymptotic behavior of the composition of two resolvents, Nonlinear Analysis: Theory, Methods, and Applications, vol.5, pp.283-301, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00017826

H. H. Bauschke and P. L. Combettes, The baillon-haddad theorem revisited, J. of Convex Analysis, vol.17, issue.4, pp.781-787, 2010.

H. H. Bauschke and P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00643354

A. Beck and N. Hallak, On the minimization over sparse symmetric sets: Projections, optimality conditions, and algorithms, 2015.

T. Blumensath, Compressed sensing with nonlinear observations and related nonlinear optimization problems, IEEE Transactions on Information Theory, vol.59, issue.6, pp.3466-3474, 2013.

T. Blumensath, M. E. Davies, C. Kervrann, P. Bouthemy, P. Elbau et al., Patch-based nonlocal functional for denoising fluorescence microscopy image sequences, IEEE Trans. Med. Imaging, vol.29, issue.2, pp.442-454, 2008.

E. Candès, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. Information Theory, IEEE Trans. on, vol.52, issue.2, pp.489-509, 2006.

A. Cegielski, Iterative methods for fixed point problems in Hilbert spaces, 2013.

G. Chierchia, N. Pustelnik, J. Pesquet, and B. Pesquet-popescu, Epigraphical projection and proximal tools for solving constrained convex optimization problems: Part i, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00744603

P. L. Combettes and J. Pesquet, A Douglas-Rachford splittting approach to nonsmooth convex variational signal recovery, IEEE J. Selec. Top. Sig. Pro, vol.1, issue.4, pp.564-574, 2007.

P. L. Combettes and J. Pesquet, Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued and variational analysis, vol.20, pp.307-330, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00794044

W. Dai and O. Milenkovic, Subspace pursuit for compressive sensing signal reconstruction, IEEE Transactions on Information Theory, vol.55, issue.5, pp.2230-2249, 2009.

S. Foucart, Hard thresholding pursuit: an algorithm for compressive sensing, SIAM Journal on Numerical Analysis, vol.49, issue.6, pp.2543-2563, 2011.

P. Jain, A. Tewari, I. S. Dhillon, J. Shawe-taylor, R. Zemel et al., Orthogonal matching pursuit with replacement, Advances in Neural Information Processing Systems 24, pp.1215-1223, 2011.

P. Jain, A. Tewari, and P. Kar, On iterative hard thresholding methods for high-dimensional m-estimation, Advances in Neural Information Processing Systems, pp.685-693, 2014.

A. Jalali, C. C. Johnson, and P. K. Ravikumar, On learning discrete graphical models using greedy methods, Advances in Neural Information Processing Systems 24, pp.1935-1943, 2011.

A. Jones, A. Tamtögl, I. Calvo-almazán, and A. Hansen, Continuous compressed sensing for surface dynamical processes with helium atom scattering, Scientific reports, vol.6, p.27776, 2016.

S. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, vol.48, 2011.

C. Lemaréchal and C. Sagastizábal, Practical Aspects of the Moreau-Yosida Regularization: Theoretical Preliminaries, SIAM J. on Optimization, vol.7, issue.2, pp.367-385, 1997.

M. Makitalo and A. Foi, Optimal inversion of the anscombe transformation in low-count poisson image denoising. Image Processing, IEEE Trans. on, vol.20, issue.1, pp.99-109, 2011.

S. G. Mallat and Z. Zhang, Matching pursuits with time-frequency dictionaries. Signal Processing, IEEE Trans. on, vol.41, issue.12, pp.3397-3415, 1993.

D. Needell and J. A. Tropp, CoSaMP: Iterative signal recovery from incomplete and inaccurate samples, Applied and Computational Harmonic Analysis, vol.26, issue.3, pp.301-321, 2009.

G. Peyré and J. Fadili, Group sparsity with overlapping partition functions, Signal Processing Conference, pp.303-307, 2011.

S. Shalev-shwartz, N. Srebro, and T. Zhang, Trading accuracy for sparsity in optimization problems with sparsity constraints, SIAM Journal on Optimization, vol.20, issue.6, pp.2807-2832, 2010.

V. N. Temlyakov, Greedy approximation, Acta Numerica, vol.17, pp.235-409, 2008.

J. A. Tropp, Greed is good: Algorithmic results for sparse approximation, IEEE Trans. Inf. Theor, vol.50, issue.10, pp.2231-2242, 2006.

S. Vaiter, C. Deledalle, G. Peyré, C. Dossal, and J. Fadili, Local behavior of sparse analysis regularization: Applications to risk estimation, Applied and Computational Harmonic Analysis, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00687751

Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, Image quality assessment: From error visibility to structural similarity. Image Processing, IEEE Trans. on, vol.13, issue.4, 2004.

Z. Yang, Z. Wang, H. Liu, Y. C. Eldar, and T. Zhang, Sparse nonlinear regression: Parameter estimation under nonconvexity, Proceedings of the 33nd International Conference on Machine Learning, pp.2472-2481, 2016.

X. Yuan, P. Li, and T. Zhang, Gradient hard thresholding pursuit for sparsity-constrained optimization, The 31st International Conference on Machine Learning, pp.127-135, 2014.

B. Zhang, J. M. Fadili, and J. Starck, Wavelets, ridgelets, and curvelets for Poisson noise removal. Image Processing, IEEE Trans. on, vol.17, issue.7, pp.1093-1108, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00259509

T. Zhang, Sparse recovery with orthogonal matching pursuit under RIP. Information Theory, IEEE Trans. on, vol.57, issue.9, pp.6215-6221, 2011.