期刊
2016 24TH EUROPEAN SIGNAL PROCESSING CONFERENCE (EUSIPCO)
卷 -, 期 -, 页码 1267-1271出版社
IEEE
关键词
Compressed sensing; structured matrices; replica method; Basis Pursuit
资金
- National Science Foundation INSPIRE (track 1) [1344069]
- Division Of Physics
- Direct For Mathematical & Physical Scien [1344069] Funding Source: National Science Foundation
Sparse reconstruction algorithms aim to retrieve high-dimensional sparse signals from a limited number of measurements. A common example is LASSO or Basis Pursuit where sparsity is enforced using an l(1)-penalty together with a cost function parallel to y - Hx parallel to (2)(2). For random design matrices H, a sharp phase transition boundary separates the ` good' parameter region where error-free recovery of a sufficiently sparse signal is possible and a `bad' regime where the recovery fails. However, theoretical analysis of phase transition boundary of the correlated variables case lags behind that of uncorrelated variables. Here we use replica trick from statistical physics to show that when an N - dimensional signal x is K-sparse and H is M x N dimensional with the covariance E [H-ia H-jb] = 1/M C-ij D-ab, with all D-aa = 1, the perfect recovery occurs at M similar to psi k (D) K log (N / M) in the very sparse limit, where psi K (D) >= 1, indicating need for more observations for the same degree of sparsity.
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