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

A Decoding Approach to Fault Tolerant Control of Linear Systems with Quantized Disturbance Input

The aim of this paper is to propose an alternative method to solve a Fault Tolerant Control problem. The model is a linear system affected by a disturbance term: this represents a large class of technological faulty processes. The goal is to make the system able to tolerate the undesired perturbation, i.e., to remove or at least reduce its negative effects; such a task is performed in three steps: the detection of the fault, its identification and the consequent process recovery. When the disturbance function is known to be \emph{quantized} over a finite number of levels, the detection can be successfully executed by a recursive \emph{decoding} algorithm, arising from Information and Coding Theory and suitably adapted to the control framework. This technique is analyzed and tested in a flight control issue; both theoretical considerations and simulations are reported

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

Deconvolution of Quantized-Input Linear Systems: An Information-Theoretic Approach

The deconvolution problem has been drawing the attention of mathematicians, physicists and engineers since the early sixties. Ubiquitous in the applications, it consists in recovering the unknown input of a convolution system from noisy measurements of the output. It is a typical instance of inverse, ill-posed problem: the existence and uniqueness of the solution are not assured and even small perturbations in the data may cause large deviations in the solution. In the last fifty years, a large amount of estimation techniques have been proposed by different research communities to tackle deconvolution, each technique being related to a peculiar engineering application or mathematical set. In many occurrences, the unknown input presents some known features, which can be exploited to develop ad hoc algorithms. For example, prior information about regularity and smoothness of the input function are often considered, as well as the knowledge of a probabilistic distribution on the input source: the estimation techniques arising in different scenarios are strongly diverse. Less effort has been dedicated to the case where the input is known to be affected by discontinuities and switches, which is becoming an important issue in modern technologies. In fact, quantized signals, that is, piecewise constant functions that can assume only a finite number of values, are nowadays widespread in the applications, given the ongoing process of digitization concerning most of information and communication systems. Moreover, hybrid systems are often encountered, which are characterized by the introduction of quantized signals into physical, analog communication channels. Motivated by such consideration, this dissertation is devoted to the study of the deconvolution of continuous systems with quantized input; in particular, our attention will be focused on linear systems. Given the discrete nature of the input, we will show that the whole problem can be interpreted as a paradigmatic digital transmission problem and we will undertake an Information-theoretic approach to tackle it. The aim of this dissertation is to develop suitable deconvolution algorithms for quantized-input linear systems, which will be derived from known decoding procedures, and to test them in different scenarios. Much consideration will be given to the theoretical analysis of these algorithms, whose performance will be rigorously described in mathematical terms

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