A game theoretic approach to expander-based compressive sensing

We consider the following expander-based compressive sensing (e-CS) problem: Given Φ ∈ ℝ^M×N (M < N), which is the adjacency matrix of an expander graph, and a vector y ∈ ℝ^M, we seek to find a vector x* with at most k-nonzero entries such that x* = argmin ∥x∥o≤κ∥y-Φ∥1, whenever it exists (k≪N). Such problems are not only nonsmooth, barring naive convexified sparse recovery approaches, but also are NP-Hard in general. To handle the non-smoothness, we provide a saddle-point reformulation of the e-CS problem, and propose a novel approximation scheme, called the game-theoretic approximate matching estimator (GAME) algorithm. We then show that the restricted isometry property of expander matrices in the ℓ₁-norm circumvents the intractability of e-CS in the worst case. GAME therefore finds a sparse approximation x to optimal solution such that ∥x^*- x̂∥ = O (∥y - Φx ∥x^*1). We also propose a convex optimization approach to e-CS based on Nesterov smoothing, and discuss its (dis)advantages.