On recovery of sparse signals via l1 minimization

Authors:
T. Tony Cai;Guangwu Xu;Jun Zhang
Affiliations:
Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA;Department of Electrical Engineering and Computer Science, University of Wisconsin-Milwaukee, Milwaukee, WI;Department of Electrical Engineering and Computer Science, University of Wisconsin-Milwaukee, Milwaukee, WI
Venue:
IEEE Transactions on Information Theory
Year:
2009

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Abstract

This paper considers constrained l1 minimization methods in a unified framework for the recovery of high-dimensional sparse signals in three settings: noiseless, bounded error, and Gaussian noise. Both l1 minimization with an l∞ constraint (Dantzig selector) and l1 minimization under an l2 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 l1 minimization with an l2 constraint and l1 minimization with an l∞ 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.