Successive Concave Sparsity Approximation for Compressed Sensing

Abstract

Abstract-In this paper, based on a successively accuracyincreasing approximation of the 0 norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the 0 norm can be controlled. We prove that the series of the approximations asymptotically coincides with the 1 and 0 norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed which leads to a number of weighted 1 minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.

Sparse Signal Recovery Using Iterative Proximal Projection

“…Moreover, as shown in [32], f scsa σ provides a tighter approximation to the 0 norm than the SL0 function. SCSA solves the following problem by employing the proximal algorithms:…”

“…1. With this illustration in mind, it seems that for each particular value of σ, the SCSA penalty f scsa σ approximates the p norms for 0 ≤ p ≤ 1 [32]. For instance, σ = 100 and σ = 0.1 correspond to p = 1 and p = 0, respectively, while σ = 1 corresponds to some 0 < p < 1.…”

“…2) Successive Concave Sparsity Approximation: A recent algorithm, called successive concave sparsity approximation (SCSA) [32], considers the following sparsity-inducing penalty:…”

“…A closed-form solution has been derived in [32] for the proximal mapping of f scsa σ . The overall procedure for solving (19) is based on the FBS framework stated in (4).…”

“…Additionally, similar to the ISP and IPP algorithms, a decreasing sequence of σ is considered, and for each value of this sequence a few iterations of (4) are performed. As shown in [32], starting with a sufficiently large value for σ, the shrinkage, i.e., the forward step of the FBS scheme, would be soft-thresholding, and as σ decreases along the outer-loop iterations, the shrinkage moves toward hard-thresholding. This behavior is shown in Fig.…”