Date of this Version
IEEE Transactions on Signal Processing
This paper establishes new restricted isometry conditions for compressed sensing and affine rank minimization. It is shown for compressed sensing that δ<;i>kA<;/i>+θ<;i>k<;/i>,<;i>kA<;/i> <; 1 guarantees the exact recovery of all <;i>k<;/i> sparse signals in the noiseless case through the constrained <;i>l<;/i><;sub>1<;/sub> minimization. Furthermore, the upper bound 1 is sharp in the sense that for any ε > 0, the condition δ<;i>kA<;/i> + θ<;i>k<;/i>,<;i>kA<;/i> <; 1+ε is not sufficient to guarantee such exact recovery using any recovery method. Similarly, for affine rank minimization, if δ<;i>rM<;/i>+θ<;i>r<;/i>,<;i>rM<;/i> <; 1 then all matrices with rank at most <;i>r<;/i> can be reconstructed exactly in the noiseless case via the constrained nuclear norm minimization; and for any ε > 0, δ<;i>rM<;/i> +θ<;i>r<;/i>,<;i>rM<;/i> <; 1+ε does not ensure such exact recovery using any method. Moreover, in the noisy case the conditions δ<;i>kA<;/i>+θ<;i>k<;/i>,<;i>kA<;/i> <; 1 and δ<;i>rM<;/i>+θ<;i>r<;/i>,<;i>rM<;/i> <; 1 are also sufficient for the stable recovery of sparse signals and low-rank matrices respectively. Applications and extensions are also discussed.
compressed sensing, matrix algebra, minimization, affine rank minimization, compressed sensing, constrained l1 minimization, constrained nuclear norm minimization, low-rank matrices, noiseless case, recovery method, restricted isometry conditions, sparse signals, compressed sensing, image reconstruction, noise measurement, signal processing, sparse matrices, vectors, Dantzig selector, compressed sensing, constrained nuclear norm minimization, low-rank matrix recovery, restricted isometry, sparse signal recovery
Cai, T., & Zhang, A. (2013). Compressed Sensing and Affine Rank Minimization Under Restricted Isometry. IEEE Transactions on Signal Processing, 61 (13), 3279-3290. http://dx.doi.org/10.1109/TSP.2013.2259164
Date Posted: 27 November 2017
This document has been peer reviewed.