Stability of l1 minimisation in compressed sensing

We discuss known results (c.f. [16, 6]) about stability of `1 minimisation (denoted ∆1) with respect to the measurement error and how those results depend on the measurement matrix Φ. Then we produce a large class of measurement matrices Φ for which we can apply results from [16] so we have estimate ‖∆1(Φ(x) + r)− x‖2 ≤ C(‖r‖2 + k−1/2σ1 k(x)). (1) We conclude with a modification of `1 minimisation which gives (1) for most random measurement matrices considered in compressed sensing literature. We also discuss stability of instance optimality in probability. 1 General description of compressed sensing Let us start by explaining the general setup of compressed sensing. Given N >> n we look for n × N a matrix Φ such that vector y = Φx ∈ R preserves information about x ∈ R . We need a decoder (generally nonlinear) ∆ : R → R such that ∆(Φx) looks like x. We require a k so ∆(Φx) = x for x any k-sparse vector. We want ∆ to be numerically friendly and k big. This leads to requiring that Φ has RIP(k, δ). Definition 1.1 ([3]) Matrix Φ has RIP(k, δ), 0 < δ < 1 if (1− δ)‖x‖2 ≤ ‖Φx‖2 ≤ (1 + δ)‖x‖2 for every k sparse vector x ∈ R . The largest possible k is ∼ n/ log(N/n). All matrices which perform well for this maximal range of k are random. In this paper we consider only random matrices Φ(ω) = (φi,j(ω)) where φi,j ’s are independent, i.i.d. subgaussian random variables e.g. Gaussian, Bernoulli. We normalize E|φi,j | = 1/n so columns Φj of Φ have typically norm one. Let us call them standard matrices. Given 0 < δ < 1 standard Φ(ω) satisfies RIP(k, δ) for k = bc1(δ)n/ log(N/n)c with probability ≥ 1 − e−c2(δ)n where c1, c2 > 0. This is well known, see e.g [7, 3, 1, 11]. The important point is that RIP is not practically verifiable even for moderately large N and k. There are two main approaches to finding ∆ for the above matrices. One approach, introduced by (E. Candes, D.Donoho et. al. see e.g. [3, 7]) is `1 minimization ∆1 i.e. ∆1(y) = Argmin{‖z‖1 : Φ(z) = y}. (2) Another uses greedy algorithm. We start with an algorithm AL which for y ∈ R and vectors (Φj)j=1 gives a subset Λ ⊂ {1, . . . , N} with #Λ ≤ k and