ℓ2/ℓ2-Foreach Sparse Recovery with Low Risk

In this paper, we consider the "foreach" sparse recovery problem with failure probability p. The goal of the problem is to design a distribution over m ×N matrices Φ and a decoding algorithm A such that for every x∈ℝN, we have with probability at least 1−p$$\|\mathbf{x}-A(\Phi\mathbf{x})\|_2\leqslant C\|\mathbf{x}-\mathbf{x}_k\|_2,$$ where xk is the best k-sparse approximation of x. Our two main results are: (1) We prove a lower bound on m, the number measurements, of Ω(klog(n/k)+log(1/p)) for $2^{-\Theta(N)}\leqslant p <1$. Cohen, Dahmen, and DeVore [4] prove that this bound is tight. (2) We prove nearly matching upper bounds that also admit sub-linear time decoding. Previous such results were obtained only when p=Ω(1). One corollary of our result is an an extension of Gilbert et al. [6] results for information-theoretically bounded adversaries. $$|\mathbf{x}-A(\Phi\mathbf{x})\|_2\leqslant C\|\mathbf{x}-\mathbf{x}_k\|_2,$$ where xk is the best k-sparse approximation of x. Our two main results are: (1) We prove a lower bound on m, the number measurements, of Ω(klog(n/k)+log(1/p)) for $2^{-\Theta(N)}\leqslant p <1$. Cohen, Dahmen, and DeVore [4] prove that this bound is tight. (2) We prove nearly matching upper bounds that also admit sub-linear time decoding. Previous such results were obtained only when p=Ω(1). One corollary of our result is an an extension of Gilbert et al. [6] results for information-theoretically bounded adversaries. $$|\mathbf{x}-A(\Phi\mathbf{x})\|_2\leqslant C\|\mathbf{x}-\mathbf{x}_k\|_2,$$ where xk is the best k-sparse approximation of x. Our two main results are: (1) We prove a lower bound on m, the number measurements, of Ω(klog(n/k)+log(1/p)) for $2^{-\Theta(N)}\leqslant p <1$. Cohen, Dahmen, and DeVore [4] prove that this bound is tight. (2) We prove nearly matching upper bounds that also admit sub-linear time decoding. Previous such results were obtained only when p=Ω(1). One corollary of our result is an an extension of Gilbert et al. [6] results for information-theoretically bounded adversaries. $$|\mathbf{x}-A(\Phi\mathbf{x})\|_2\leqslant C\|\mathbf{x}-\mathbf{x}_k\|_2,$$ where xk is the best k-sparse approximation of x. Our two main results are: (1) We prove a lower bound on m, the number measurements, of Ω(klog(n/k)+log(1/p)) for $2^{-\Theta(N)}\leqslant p <1$. Cohen, Dahmen, and DeVore [4] prove that this bound is tight. (2) We prove nearly matching upper bounds that also admit sub-linear time decoding. Previous such results were obtained only when p=Ω(1). One corollary of our result is an an extension of Gilbert et al. [6] results for information-theoretically bounded adversaries.