Abstract
The recently proposed modified-compressive sensing (modified-CS), which utilizes the partially known support as prior knowledge, significantly improves the performance of recovering sparse signals. However, modified-CS depends heavily on the reliability of the known support. An important problem, which must be studied further, is the recoverability of modified-CS when the known support contains a number of errors. In this letter, we analyze the recoverability of modified-CS in a stochastic framework. A sufficient and necessary condition is established for exact recovery of a sparse signal. Utilizing this condition, the recovery probability that reflects the recoverability of modified-CS can be computed explicitly for a sparse signal with nonzero entries. Simulation experiments have been carried out to validate our theoretical results.
Highlights
A central problem in CS is the following: given an m|n matrix A, and a measurement vector y~AxÃ, recover xÃ
Vaswani and Lu [6,7,8,9], Miosso [10,11], Wang and Yin [12,13], Friedlander et al [14], Jacques [15] have shown that exact recovery based on fewer measurements than those needed for the ‘1-minimization approach is possible when the support of xà is partially known
The set operations | and \ stand for set union and set difference respectively
Summary
A central problem in CS is the following: given an m|n matrix A (mvn), and a measurement vector y~AxÃ, recover xÃ. To deal with this problem, the most extensively studied recovery method is the ‘1-minimization approach (Basis Pursuit) [1,2,3,4,5]. This convex problem can be solved efficiently; O(‘ log(n=‘)) probabilistic measurements are sufficient for it to recover a ‘-sparse vector xà (i.e., all but at most ‘ entries are zero) exactly. Since the support evolve slowly over time, the previously recovered support can be used as known part for later reconstruction
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