An ALPS view of sparse recovery

We provide two compressive sensing (CS) recovery algorithms based on iterative hard-thresholding. The algorithms, collectively dubbed as algebraic pursuits (ALPS), exploit the restricted isometry properties of the CS measurement matrix within the algebra of Nesterov's optimal gradient methods. We theoretically characterize the approximation guarantees of ALPS for signals that are sparse on ortho-bases as well as on tight-frames. Simulation results demonstrate a great potential for ALPS in terms of phase-transition, noise robustness, and CS reconstruction.

[1]  Simon Foucart,et al.  Hard Thresholding Pursuit: An Algorithm for Compressive Sensing , 2011, SIAM J. Numer. Anal..

[2]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

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

[4]  Volkan Cevher,et al.  Low-Dimensional Models for Dimensionality Reduction and Signal Recovery: A Geometric Perspective , 2010, Proceedings of the IEEE.

[5]  Volkan Cevher,et al.  On accelerated hard thresholding methods for sparse approximation , 2011, Optical Engineering + Applications.

[6]  Yonina C. Eldar,et al.  Compressed Sensing with Coherent and Redundant Dictionaries , 2010, ArXiv.

[7]  David L. Donoho,et al.  Precise Undersampling Theorems , 2010, Proceedings of the IEEE.

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

[9]  Rahul Garg,et al.  Gradient descent with sparsification: an iterative algorithm for sparse recovery with restricted isometry property , 2009, ICML '09.

[10]  Y. Nesterov.  A method for solving the convex programming problem with convergence rate O(1/k^2) , 1983 .

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

[12]  Pierre Vandergheynst,et al.  Compressed Sensing and Redundant Dictionaries , 2007, IEEE Transactions on Information Theory.

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

[14]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..