Denoising based Vector Approximate Message Passing

The D-AMP methodology, recently proposed by Metzler, Maleki, and Baraniuk, allows one to plug in sophisticated denoisers like BM3D into the AMP algorithm to achieve state-of-the-art compressive image recovery. But AMP diverges with small deviations from the i.i.d.-Gaussian assumption on the measurement matrix. Recently, the VAMP algorithm has been proposed to fix this problem. In this work, we show that the benefits of VAMP extend to D-VAMP. Consider the problem of recovering a (vectorized) image x0 ∈ R from compressive (i.e., M ≪ N ) noisy linear measurements y = Φx0 +w ∈ R M , (1) known as “compressive imaging.” The “sparse” approach to this problem exploits sparsity in the coefficients v0 , Ψx0 ∈ R N of an orthonormal wavelet transform Ψ. The idea is to rewrite (1) as y = Av0 +w for A , ΦΨ , (2) recover an estimate v̂ of v0 from y, and then construct the image estimate as x̂ = Ψ v̂. Although many algorithms have been proposed for sparse recovery of v0, a notable one is the approximate message passing (AMP) algorithm from [1]. It is computationally efficient (i.e., one multiplication by A and A per iteration and relatively few iterations) and its performance, when M and N are large and Φ is zero-mean i.i.d. Gaussian, is rigorously characterized by a scalar state evolution. A variant called “denoising-based AMP” (D-AMP) was recently proposed [2] for direct recovery of x0 from (1). It exploits the fact that, at iteration t, AMP constructs a pseudo-measurement of the form v0 + N (0, σ t I) with known σ t , which is amenable to any image denoising algorithm. By plugging in a state-of-the-art image denoiser like BM3D [3], D-AMP yields state-of-the-art compressive imaging. AMP and D-AMP, however, have a serious weakness: they diverge under small deviations from the zero-mean i.i.d. Gaussian assumption on Φ, such as non-zero mean or mild ill-conditioning. A robust alternative called “vector AMP” (VAMP) was recently proposed [4]. VAMP has similar complexity to AMP and a rigorous state evolution November 7, 2016 DRAFT