Revisiting compressed sensing: exploiting the efficiency of simplex and sparsification methods

Abstract

We propose two approaches to solve large-scale compressed sensing problems. The first approach uses the parametric simplex method to recover very sparse signals by taking a small number of simplex pivots, while the second approach reformulates the problem using Kronecker products to achieve faster computation via a sparser problem formulation. In particular, we focus on the computational aspects of these methods in compressed sensing. For the first approach, if the true signal is very sparse and we initialize our solution to be the zero vector, then a customized parametric simplex method usually takes a small number of iterations to converge. Our numerical studies show that this approach is 10 times faster than state-of-the-art methods for recovering very sparse signals. The second approach can be used when the sensing matrix is the Kronecker product of two smaller matrices. We show that the best-known sufficient condition for the Kronecker compressed sensing (KCS) strategy to obtain a perfect recovery is more restrictive than the corresponding condition if using the first approach. However, KCS can be formulated as a linear program with a very sparse constraint matrix, whereas the first approach involves a completely dense constraint matrix. Hence, algorithms that benefit from sparse problem representation, such as interior point methods (IPMs), are expected to have computational advantages for the KCS problem. We numerically demonstrate that KCS combined with IPMs is up to 10The first author's research is supported by ONR Award N00014-13-1-0093, the third author's by NSF Grant III-1116730, and the fourth author's by NSF Grant DMS-1005539.

On the spectral structure of Jordan-Kronecker products of symmetric and skew-symmetric matrices

“…Kronecker products have many applications in signal processing, semidefinite programming, and quantum computing. They have also been used extensively in the theory and applications of linear matrix equations such as Sylvester equations and Lyapunov problem [21], in some compressed sensing applications using sparsification [23] and in constructing convex relaxations of non-convex sets [1]. Tensor product preconditioners used in conjugate gradient method [20] and image restoration [15], low rank tensor approximations arising in the context of certain quantum chemistry systems [22], quantum many-body systems [12], and high dimensional partial differential equations [2,3] are among many other applications which utilize the rich properties of tensor products that can transfer the structure of the individual elements to the product itself.…”