Instance-optimality in Probability with an `1-Minimization Decoder

Let Φ(ω), ω ∈ Ω, be a family of n ×N random matrices whose entries φi,j are independent realizations of a symmetric, real random variable η with expectation IEη = 0 and variance IEη2 = 1/n. Such matrices are used in compressed sensing to encode a vector x ∈ IR by y = Φx. The information y holds about x is extracted by using a decoder ∆ : IR → IR . The most prominent decoder is the `1-minimization decoder ∆ which gives for a given y ∈ IR the element ∆(y) ∈ IR which has minimal `1-norm among all z ∈ IR with Φz = y. This paper is interested in properties of the random family Φ(ω) which guarantee that the vector x̄ := ∆(Φx) will with high probability approximate x in `2 to an accuracy comparable with the best k-term error of approximation in `2 for the range k ≤ an/ log2(N/n). This means that for the above range of k, for each signal x ∈ IR , the vector x̄ := ∆(Φx) satisfies ‖x− x̄‖`N2 ≤ C inf z∈Σk ‖x− z‖`N2 with high probability on the draw of Φ. Here, Σk consists of all vectors with at most k nonzero coordinates. The first result of this type was proved by Wojtaszczyk [19] who showed this property when η is a normalized Gaussian random variable. We extend this property to more general random variables, including the particular case where η is the Bernoulli random variable which takes the values ±1/ √ n with equal probability. The proofs of our results use geometric mapping properties of such random matrices some of which were recently obtained in [14].