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 equation 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 x̂ to optimal solution such that ||x* - x̂ ||₁ = O (||y - Φx*||₁). We also propose a convex optimization approach to e-CS based on Nesterov smoothing, and discuss its (dis)advantages.