Maximin Analysis of Message Passing Algorithms for Recovering Block Sparse Signals

We consider the problem of recovering a block (or group) sparse signal from an underdetermined set of random linear measurements, which appear in compressed sensing applications such as radar and imaging. Recent results of Donoho, Johnstone, and Montanari have shown that approximate message passing (AMP) in combination with Stein’s shrinkage outperforms group LASSO for large block sizes. In this paper, we prove that for a fixed block size and in the strong undersampling regime (i.e., having very few measurements compared to the ambient dimension), AMP cannot improve upon group LASSO, thereby complementing the results of Donoho et al. I. PROBLEM STATEMENT We analyze the recovery of a block (or group) sparse signal x ∈ R with at most k nonzero entries from an underdetermined set of linear measurements y = Ax, where A ∈ Rn×N is i.i.d zeromean Gaussian with unit variance. We consider the asymptotic setting where δ = n/N and ρ = k/n represent the undersampling and sparsity parameters, respectively, and N,n, k →∞; we furthermore assume that all blocks are of the same size B. The signal x is partitioned into M blocks with N =MB. In what follows, we will denote a particular block by xB . Suppose that the elements of xB are drawn from a distribution F (xB) = (1 − )δ0(‖xB‖2) + G(xB), where = ρδ, and δ0 is the Dirac delta function; G is a probability distribution that is typically unknown in practice. We define the block soft-thresholding function as follows [1]: η(y; τ) , yB/‖yB‖2 max { ‖yB‖2 − τ, 0 } . (1) Two popular algorithms for recovering block sparse signals in compressed sensing are group LASSO [2] and approximate message passing (AMP) [3]. Group LASSO searches for a vector x that solves x , argminx{ ∑M B=1 ‖xB‖2 : y = Ax}. AMP is an iterative algorithm for computing x. Concretely, AMP is initialized by x = 0 and z = 0, and iteratively performs the following computations [4]: x = η(x +Ax) and z = y −Ax + c. Here, c is a correction term that depends on the previous iterations to significantly improve the convergence of AMP; x is the sparse estimate at iteration t, and η is a nonlinear function that imposes (block) sparsity. The performance of compressed sensing recovery algorithms can be characterized accurately by their phase-transition (PT) behavior. Specifically, we define a two-dimensional phase space (δ, ρ) ∈ [0, 1] that is partitioned into two regions: “success” and “failure”, with these regions separated by the PT curve (δ, ρ(δ)). For the same value of δ, algorithms with higher PT outperform algorithms with lower PT, i.e., guarantee the exact recovery for more nonzero entries k.