Abstract

We consider the following signal recovery problem: given a measurement matrix Φ ∈ and a noisy observation vector c ∈ constructed from c =  where  is the noise vector whose entries follow i.i.d. centered sub-Gaussian distribution, how to recover the signal if D is sparse under a linear transformation One natural method using convex optimization is to solve the following problem:

This paper provides an upper bound of the estimate error and shows the consistency property of this method by assuming that the design matrix Φ is a Gaussian random matrix. Specifically, we show 1) in the noiseless case, if the condition number of D is bounded and the measurement number where s is the sparsity number, then the true solution can be recovered with high probability; and 2) in the noisy case, if the condition number of D is bounded and the measurement increases faster than s log(p), that is, s log(p) = o(n), the estimate error converges to zero with probability 1 when p and s go to infinity. Our results are consistent with those for the special case (equivalently LASSO) and improve the existing analysis. The condition number of D plays a critical role in our analysis. We consider the condition numbers in two cases including the fused LASSO and the random graph: the condition number in the