Sparse Recovery Algorithms: Sufficient Conditions in Terms of Restricted Isometry Constants
We review three recovery algorithms used in Compressive Sensing for the reconstruction s-sparse vectors x∈ℂ N from the mere knowledge of linear measurements y=A x∈ℂ m , m<N. For each of the algorithms, we derive improved conditions on the restricted isometry constants of the measurement matrix A that guarantee the success of the reconstruction. These conditions are δ2s <0.4652 for basis pursuit, δ3s <0.5 and δ2s <0.25 for iterative hard thresholding, and δ4s <0.3843 for compressive sampling matching pursuit. The arguments also applies to almost sparse vectors and corrupted measurements. The analysis of iterative hard thresholding is surprisingly simple. The analysis of basis pursuit features a new inequality that encompasses several inequalities encountered in Compressive Sensing.
KeywordsCompressive Sensing Convex Polytope Basis Pursuit Recovery Algorithm Restricted Isometry Property
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The author thanks the meeting organizers, Mike Neamtu and Larry Schumaker, for welcoming a minisymposium on Compressive Sensing at the Approximation Theory conference. He also acknowledges support from the French National Research Agency (ANR) through project ECHANGE (ANR-08-EMER-006).
- 3.Candès, E., Tao. T.: Decoding by linear programing. IEEE Trans. Inf. Theory 51, 4203–4215 (2005).Google Scholar
- 5.Foucart, S.: A note on guaranteed sparse recovery via ℓ 1-minimization. Applied and Comput. Harmonic Analysis, To appear. Appl. Comput. Harmon. Anal. 29, 97–103 (2010).Google Scholar
- 6.Garg, R., Khandekar, R.: Gradient descent with sparsification: An iterative algorithm for sparse recovery with restricted isometry property. In: Bottou, L., Littman, M. (eds.) Proceedings of the 26 th International Confer- ence on Machine Learning, pp. 337-344.Google Scholar