Guaranteed sparse recovery under linear transformation

We consider the following signal recovery problem: given a measurement matrix Φ ∈ ℝ^nxp and a noisy observation vector c ∈ ℝⁿconstructed from c = Φθ*+ ε where ε ∈ ℝⁿ is the noise vector whose entries follow i.i.d. centered sub-Gaussian distribution, how to recover the signal θ&z,ast; if Dθ*is sparse under a linear transformation D ∈ ℝ^mxp? One natural method using convex optimization is to solve the following problem:(Equation Presented) This paper provides an upper bound of the estimate error and shows the consistency property of this method by assuming that the design matrix $ 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 n ≥ Ω(slog(p)) 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 slog(p), that is, slog(p) = o(n), the estimate error converges to zero with probability 1 when p and s go to infinity. Our resuits are consistent with those for the special case D = I_pxp (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 fused LASSO case is bounded by a constant, while the condition number in the random graph case is bounded with high probability ^{if m/p (i.e., #edge/#vertex) is larger than a} certain constant. Numerical simulations are consistent with our theoretical results.