16 research outputs found
Discrete Signal Reconstruction by Sum of Absolute Values
In this letter, we consider a problem of reconstructing an unknown discrete
signal taking values in a finite alphabet from incomplete linear measurements.
The difficulty of this problem is that the computational complexity of the
reconstruction is exponential as it is. To overcome this difficulty, we extend
the idea of compressed sensing, and propose to solve the problem by minimizing
the sum of weighted absolute values. We assume that the probability
distribution defined on an alphabet is known, and formulate the reconstruction
problem as linear programming. Examples are shown to illustrate that the
proposed method is effective.Comment: IEEE Signal Processing Letters (to appear
Non-convex approach to binary compressed sensing
We propose a new approach to the recovery of binary signals in compressed
sensing, based on the local minimization of a non-convex cost functional. The
desired signal is proved to be a local minimum of the functional under mild
conditions on the sensing matrix and on the number of measurements. We develop
a procedure to achieve the desired local minimum, and, finally, we propose
numerical experiments that show the improvement obtained by the proposed
approach with respect to the classical convex approach, i.e., Lasso
Recovery of binary sparse signals from compressed linear measurements via polynomial optimization
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
Advances in the recovery of binary sparse signals
Recently, the recovery of binary sparse signals from compressed linear systems has received attention due to its several applications. In this contribution, we review the latest results in this framework, that are based on a suitable non-convex polynomial formulation of the
problem. Moreover, we propose novel theoretical results. Then, we show numerical results that highlight the enhancement obtained through the non-convex approach with respect to the state-of-the-art methods
Non-convex approach to binary compressed sensing
We propose a new approach for the recovery of binary signals in compressed sensing, based on the local minimization of a non-convex cost functional. The desired signal is proved to be a local minimum of the functional under mild conditions on the sensing matrix and on the number of measurements. We develop a procedure to achieve the desired local minimum, and, finally, we propose numerical experiments that show the improvement obtained by the proposed approach with respect to classical convex methods
Efficient recovery algorithm for discrete valued sparse signals using an ADMM approach
Motivated by applications in wireless communications, in this paper we propose a reconstruction algorithm for sparse signals whose values are taken from a discrete set, using a limited number of noisy observations. Unlike conventional compressed sensing algorithms, the proposed approach incorporates knowledge of the discrete valued nature of the signal in the detection process. This is accomplished through the alternating direction method of the multipliers which is applied as a heuristic to decompose the associated maximum likelihood detection problem in order to find candidate solutions with a low computational complexity order. Numerical results in different scenarios show that the proposed algorithm is capable of achieving very competitive recovery error rates when compared with other existing suboptimal approaches.info:eu-repo/semantics/publishedVersio
An algebraic perspective on integer sparse recovery
Compressed sensing is a relatively new mathematical paradigm that shows a
small number of linear measurements are enough to efficiently reconstruct a
large dimensional signal under the assumption the signal is sparse.
Applications for this technology are ubiquitous, ranging from wireless
communications to medical imaging, and there is now a solid foundation of
mathematical theory and algorithms to robustly and efficiently reconstruct such
signals. However, in many of these applications, the signals of interest do not
only have a sparse representation, but have other structure such as
lattice-valued coefficients. While there has been a small amount of work in
this setting, it is still not very well understood how such extra information
can be utilized during sampling and reconstruction. Here, we explore the
problem of integer sparse reconstruction, lending insight into when this
knowledge can be useful, and what types of sampling designs lead to robust
reconstruction guarantees. We use a combination of combinatorial, probabilistic
and number-theoretic methods to discuss existence and some constructions of
such sensing matrices with concrete examples. We also prove sparse versions of
Minkowski's Convex Body and Linear Forms theorems that exhibit some limitations
of this framework