Recovery of Binary Sparse Signals From Compressed Linear Measurements via Polynomial Optimization

Abstract

The recovery of signals with finite-valued components from few linear measurements is a problem with widespread applications and interesting mathematical characteristics. In the compressed sensing framework, tailored methods have been recently proposed to deal with the case of finite-valued sparse signals. In this work, we focus on binary sparse signals and we propose a novel formulation, based on polynomial optimization. This approach is analyzed and compared to the state-of-the-art binary compressed sensing methods.

No Relaxation: Guaranteed Recovery of Finite-Valued Signals from Undersampled Measurements

“…Theorem 1 shows that it is possible to downsample the output of the filter h by a factor of L and yet uniquely identify a finite-valued input to the filter. In contrast to existing approaches [13,12,7], this show that the finite valued constraint alone ensures injectivity of the overall map and it is not necessary to impose any additional sparsity constraint.…”

“…A special case of this model involves binary valued signals (i.e. D " 1) and such binary valued signals, shapes or images have been considered in [7,12,13]. However, they relax the binary constraint to recover x using convex optimization, and theoretical guarantees are limited to random sampling.…”

“…The recovery of finite-valued signal x from undersampled measurements is potentially an ill posed problem when M ăă N . Most existing approaches utilize sparsity constraints and tools from compressed sensing to solve this problem [11,12,7,5,13]. However, these approaches end up relaxing the finite-valued (binary) constraint to develop convex relaxations of the original non-convex problem.…”

“…On the other hand, several lines of work on Binary Compressed Sensing [11,12,13,14] have developed modifications to the classical compressed sensing techniques to incorporate finite-value constraints on the signal in addition to sparsity. However, theoretical guarantees for these algorithms require a random matrix [11,13] or a specially designed measurement matrix [15].…”

Sparse linear regression with compressed and low-precision data via concave quadratic programming

“…In the first one, we consider the ideal setting where the true vector of parameters is x ∈ {0, d} n ; thus, in this case, we expect to recover exactly both the support and the non-zero values, under suitable conditions. We specify that, despite its peculiarity, the binary setting is relevant in widespread applications, such as localization and image processing, see [8,10] for a complete overview. In the second experiment, instead, the non-zero components are in [α, β], and d = α+β 2 ; in this case, we expect to obtain the right support and a biased estimation of the non-zero values.…”