Sparse Signal Recovery Using Iterative Proximal Projection

This paper is concerned with designing efficient algorithms for recovering sparse signals from noisy underdetermined measurements. More precisely, we consider minimization of a nonsmooth and nonconvex sparsity promoting function subject to an error constraint. To solve this problem, we use an alternating minimization penalty method, which ends up with an iterative proximal-projection approach. Furthermore, inspired by accelerated gradient schemes for solving convex problems, we equip the obtained algorithm with a so-called extrapolation step to boost its performance. Additionally, we prove its convergence to a critical point. Our extensive simulations on synthetic as well as real data verify that the proposed algorithm considerably outperforms some well-known and recently proposed algorithms.

[1]  S. Foucart,et al.  Sparsest solutions of underdetermined linear systems via ℓq-minimization for 0 , 2009 .

[2]  Scott T. Rickard,et al.  Comparing Measures of Sparsity , 2008, IEEE Transactions on Information Theory.

[3]  M. Lustig,et al.  Compressed Sensing MRI , 2008, IEEE Signal Processing Magazine.

[4]  Benar Fux Svaiter,et al.  Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods , 2013, Math. Program..

[5]  Trac D. Tran,et al.  ICR: Iterative Convex Refinement for Sparse Signal Recovery Using Spike and Slab Priors , 2015, ArXiv.

[6]  Heinz H. Bauschke,et al.  Fixed-Point Algorithms for Inverse Problems in Science and Engineering , 2011, Springer Optimization and Its Applications.

[7]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[8]  Jian Wang,et al.  Generalized Orthogonal Matching Pursuit , 2011, IEEE Transactions on Signal Processing.

[9]  Masoumeh Azghani Iterative methods for random sampling and compressed sensing recovery , 2013 .

[10]  R. Tibshirani Regression Shrinkage and Selection via the Lasso , 1996 .

[11]  Volkan Cevher,et al.  Learning with Compressible Priors , 2009, NIPS.

[12]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[13]  Jianqing Fan,et al.  Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties , 2001 .

[14]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[15]  Tong Zhang,et al.  Analysis of Multi-stage Convex Relaxation for Sparse Regularization , 2010, J. Mach. Learn. Res..

[16]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[17]  P. Tseng Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization , 2001 .

[18]  Philip Schniter,et al.  Expectation-Maximization Gaussian-Mixture Approximate Message Passing , 2012, IEEE Transactions on Signal Processing.

[19]  Stephen P. Boyd,et al.  Proximal Algorithms , 2013, Found. Trends Optim..

[20]  Marc Teboulle,et al.  Proximal alternating linearized minimization for nonconvex and nonsmooth problems , 2014, Math. Program..

[21]  慧 廣瀬 A Mathematical Introduction to Compressive Sensing , 2015 .

[22]  Hédy Attouch,et al.  On the convergence of the proximal algorithm for nonsmooth functions involving analytic features , 2008, Math. Program..

[23]  Yonina C. Eldar,et al.  Structured Compressed Sensing: From Theory to Applications , 2011, IEEE Transactions on Signal Processing.

[24]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.

[25]  Wotao Yin,et al.  A Block Coordinate Descent Method for Regularized Multiconvex Optimization with Applications to Nonnegative Tensor Factorization and Completion , 2013, SIAM J. Imaging Sci..

[26]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[27]  Stéphane Canu,et al.  Ieee Transactions on Signal Processing 1 Recovering Sparse Signals with a Certain Family of Non-convex Penalties and Dc Programming , 2022 .

[28]  Massoud Babaie-Zadeh,et al.  Successive Concave Sparsity Approximation for Compressed Sensing , 2015, IEEE Transactions on Signal Processing.

[29]  Stephen J. Wright,et al.  Computational Methods for Sparse Solution of Linear Inverse Problems , 2010, Proceedings of the IEEE.

[30]  Wotao Yin,et al.  Iteratively reweighted algorithms for compressive sensing , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[31]  Jian Zhang,et al.  Image compressive sensing recovery using adaptively learned sparsifying basis via L0 minimization , 2014, Signal Process..

[32]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[33]  Panagiotis Kosmas,et al.  Microwave Medical Imaging Based on Sparsity and an Iterative Method With Adaptive Thresholding , 2015, IEEE Transactions on Medical Imaging.

[34]  Christian Jutten,et al.  A Fast Approach for Overcomplete Sparse Decomposition Based on Smoothed $\ell ^{0}$ Norm , 2008, IEEE Transactions on Signal Processing.

[35]  Wotao Yin,et al.  A Globally Convergent Algorithm for Nonconvex Optimization Based on Block Coordinate Update , 2014, J. Sci. Comput..

[36]  Mike E. Davies,et al.  Iterative Hard Thresholding for Compressed Sensing , 2008, ArXiv.

[37]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[38]  J. Friedman Fast sparse regression and classification , 2012 .

[39]  W. H. Young On Classes of Summable Functions and their Fourier Series , 1912 .

[40]  Massoud Babaie-Zadeh,et al.  Iterative Sparsification-Projection: Fast and Robust Sparse Signal Approximation , 2016, IEEE Transactions on Signal Processing.

[41]  T. Blumensath,et al.  Theory and Applications , 2011 .