Minimization of $\ell_{1-2}$ for Compressed Sensing

Abstract

We study minimization of the difference of ℓ 1 and ℓ 2 norms as a non-convex and Lipschitz continuous metric for solving constrained and unconstrained compressed sensing problems. We establish exact (stable) sparse recovery results under a restricted isometry property (RIP) condition for the constrained problem, and a full-rank theorem of the sensing matrix restricted to the support of the sparse solution. We present an iterative method for ℓ 1−2 minimization based on the difference of convex functions algorithm (DCA), and prove that it converges to a stationary point satisfying first order optimality condition. We propose a sparsity oriented simulated annealing (SA) procedure with non-Gaussian random perturbation and prove the almost sure convergence of the combined algorithm (DCASA) to a global minimum. Computation examples on success rates of sparse solution recovery show that if the sensing matrix is ill-conditioned (non RIP satisfying), then our method is better than existing non-convex compressed sensing solvers in the literature. Likewise in the magnetic resonance imaging (MRI) phantom image recovery problem, ℓ 1−2 succeeds with 8 projections. Irrespective of the conditioning of the sensing matrix, ℓ 1−2 is better than ℓ 1 in both the sparse signal and the MRI phantom image recovery problems.To appear in SIAM J. Sci. Comput. Introduction.Compressed sensing (CS) has been a rapidly growing field of research in signal processing and mathematics stimulated by the foundational papers [8,6,22,23] and related Bregman iteration methods [57,33]. A fundamental issue in CS is to recover an ndimensional vectorx from m ≪ n measurements (the projection ofx onto m n-dimensional vectors), or in matrix form given b = Ax, where A is the so-called m × n sensing (measurements) matrix. One can also viewx as coefficients of a sparse linear representation of data b in terms of redundant columns of matrix A known as dictionary elements.The conditioning of A is related to its restricted isometry property (RIP) as well as the coherence (maximum of pairwise mutual angles) of the column vectors of A. Breakthrough results in CS have established when A is drawn from a Gaussian matrix ensemble or random row sampling without replacement from an orthogonal matrix (Fourier matrix), then A is wellconditioned in the sense that ifx is s-sparse (s is much less than n), m = O(s log n) measurements suffice to recoverx (the sparsest solution) with an overwhelming probability by ℓ 1 minimization or the basis pursuit (BP) problem [6, 15]:(1.1) minx ∥x∥ 1 subject to Ax = b.In the above formulation, ℓ 1 norm works as the convex relaxation of ℓ 0 that counts the nonzeros. Such a matrix A has incoherent column vectors. On the other hand, if columns of A are coherent enough, such as those arising in discretization of continuum imaging problems (radar and medical imaging) when the grid spacing is below the Rayleigh threshold [25], ℓ 1 minimization may not give the sparsest solution [25,55].

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

“…It is shown in [21,40] that any limit point of the DCA sequence converges to a stationary point; and in Theorem 5, we give a tighter result, which states that the limit point is dstationary rather than stationary. These stationarity concepts are related as the set of local minimizers belongs to the set of d-stationary points, which belongs to the set of stationary points.…”

“…The former is recently proposed in [22,40] as an alternative to L 1 for CS, and the latter is often used in statistics and machine learning [33,41]. Numerical simulations show that L 1 minimization often fails when MSF < 1, in which case we demonstrate that both L 1−2 and CL 1 outperform the classical L 1 method.…”

“…This method gives better results than the classical L 1 approaches in the RIP regime and/or incoherent scenario, but it does not work so well for highly coherent CS, as observed in [21,22,40].…”

“…Proof In [40], the DCA sequence {x k } was shown to converge to a stationary point x * . We now prove that all the iterates (except for the first one) and the limit point x * are non-zero.…”

s-SMOOTH: Sparsity and Smoothness Enhanced EEG Brain Tomography

“…vTGV enhances the smoothness of the brain image and reconstructs the spatial distribution of the current density more precisely. Meanwhile, motivated by the performance of the ℓ 1−2 regularization in compressive sensing reconstruction and other image processing problems (Esser et al, 2013; Lou et al, 2014; Yin et al, 2015), we incorporate the ℓ 1−2 regularization into the objective function. Numerical experiments show that ℓ 1−2 regularization provides faster convergence and yields sparser source image than the ℓ 1 -norm regularization.…”