On Recovery of Sparse Signals Via \ell Minimization

• Published in: IEEE transactions on information theory Vol. 55; no. 7; pp. 3388 - 3397
• Main Authors: Cai, T.T., Guangwu Xu, Jun Zhang
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2009; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Compressed sensing; Dantzig selector ell _{1} minimization; Equations; Gaussian noise; Information processing; Least squares methods; Linear regression; Minimization methods; Noise measurement; Optimization techniques; restricted isometry property; Signaling; sparse recovery; sparsity; Statistics; Vectors
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2009.2021377

This paper considers constrained lscr 1 minimization methods in a unified framework for the recovery of high-dimensional sparse signals in three settings: noiseless, bounded error, and Gaussian noise. Both lscr 1 minimization with an lscr infin constraint (Dantzig selector) and lscr 1 minimization under an l lscr 2 constraint are considered. The results of this paper improve the existing results in the literature by weakening the conditions and tightening the error bounds. The improvement on the conditions shows that signals with larger support can be recovered accurately. In particular, our results illustrate the relationship between lscr 1 minimization with an l lscr 2 constraint and lscr 1 minimization with an lscr infin constraint. This paper also establishes connections between restricted isometry property and the mutual incoherence property. Some results of Candes, Romberg, and Tao (2006), Candes and Tao (2007), and Donoho, Elad, and Temlyakov (2006) are extended.