During the last decades, information is being gathered and processed at an explosive rate. This fact gives rise to a very important issue, that is, how to effectively and precisely describe the information content of a given source signal or an ensemble of source signals, such that it can be stored, processed or transmitted by taking into consideration the limitations and capabilities of the several digital devices. One of the fundamental principles of signal processing for decades is the Nyquist-Shannon sampling theorem, which states that the minimum number of samples needed to reconstruct a signal without error is dictated by its bandwidth. However, there are many cases in our everyday life in which sampling at the Nyquist rate results in too many data and thus, demanding an increased processing power, as well as storage requirements. A mathematical theory that emerged ...

This thesis develops new mathematical theory and presents novel recovery algorithms for discrete linear dynamical systems (LDS) with sparsity constraints on either control inputs or initial state. The recovery problems in this framework manifest as the problem of reconstructing one or more sparse signals from a set of noisy underdetermined linear measurements. The goal of our work is to design algorithms for sparse signal recovery which can exploit the underlying structure in the measurement matrix and the unknown sparse vectors, and to analyze the impact of these structures on the efficacy of the recovery. We answer three fundamental and interconnected questions on sparse signal recovery problems that arise in the context of LDS. First, what are necessary and sufficient conditions for the existence of a sparse solution? Second, given that a sparse solution exists, what are good low-complexity algorithms that ...

Chapter 2 contains a short introduction to the fundamentals of compressed sensing theory, which is the larger context of this thesis. We start with introducing the key concepts of sparsity and sparse representations of signals. We discuss the central problem of compressed sensing, i.e. how to adequately recover sparse signals from a small number of measurements, as well as the multiple formulations of the reconstruction problem. A large part of the chapter is devoted to some of the most important conditions necessary and/or sufficient to guarantee accurate recovery. The aim is to introduce the reader to the basic results, without the burden of detailed proofs. In addition, we also present a few of the popular reconstruction and optimization algorithms that we use throughout the thesis. Chapter 3 presents an alternative sparsity model known as analysis sparsity, that offers similar recovery ...

As many machine learning and signal processing problems are fundamentally nonconvex and too expensive/difficult to be convexified, my research is focused on understanding the optimization landscapes of their fundamentally nonconvex formulations. After understanding their optimization landscapes, we can develop optimization algorithms to efficiently navigate these optimization landscapes and achieve the global optimality convergence. So, the main theme of this thesis would be optimization, with an emphasis on nonconvex optimization and algorithmic developments for these popular optimization problems. This thesis can be conceptually divided into four parts: Part 1: Convex Optimization. In the first part, we apply convex relaxations to several popular nonconvex problems in signal processing and machine learning (e.g. line spectral estimation problem and tensor decomposition problem) and prove that the solving the new convex relaxation problems is guaranteed to achieve the globally optimal solutions of their original nonconvex ...

Today’s society is characterized by an abundance of data that is generated at an unprecedented velocity. However, much of this data is immediately thrown away by compression or information extraction. In a compressed sensing (CS) setting the inherent sparsity in many datasets is exploited by avoiding the acquisition of superfluous data in the first place. We combine this technique with tensors, or multiway arrays of numerical values, which are higher-order generalizations of vectors and matrices. As the number of entries scales exponentially in the order, tensor problems are often large-scale. We show that the combination of simple, low-rank tensor decompositions with CS effectively alleviates or even breaks the so-called curse of dimensionality. After discussing the larger data fusion optimization framework for coupled and constrained tensor decompositions, we investigate three categories of CS type algorithms to deal with large-scale problems. First, ...

System identification studies how to construct mathematical models for dynamical systems from the input and output data, which finds applications in many scenarios, such as predicting future output of the system or building model based controllers for regulating the output the system. Among many other methods, convex optimization is becoming an increasingly useful tool for solving system identification problems. The reason is that many identification problems can be formulated as, or transformed into convex optimization problems. This transformation is commonly referred to as the convexification technique. The first theme of the thesis is to understand the efficacy of the convexification idea by examining two specific examples. We first establish that a l1 norm based approach can indeed help in exploiting the sparsity information of the underlying parameter vector under certain persistent excitation assumptions. After that, we analyze how the nuclear ...

The effectiveness of machine learning (ML) in today's applications largely depends on the goodness of the representation of data used within the ML algorithms. While the massiveness in dimension of modern day data often requires lower-dimensional data representations in many applications for efficient use of available computational resources, the use of uncorrelated features is also known to enhance the performance of ML algorithms. Thus, an efficient representation learning solution should focus on dimension reduction as well as uncorrelated feature extraction. Even though Principal Component Analysis (PCA) and linear autoencoders are fundamental data preprocessing tools that are largely used for dimension reduction, when engineered properly they can also be used to extract uncorrelated features. At the same time, factors like ever-increasing volume of data or inherently distributed data generation impede the use of existing centralized solutions for representation learning that require ...

This dissertation details three approaches for direction-of-arrival (DOA) estimation or beamforming in array signal processing from the perspective of sparsity. In the first part of this dissertation, we consider sparse array beamformer design based on the alternating direction method of multipliers (ADMM); in the second part of this dissertation, the problem of joint DOA estimation and distorted sensor detection is investigated; and off-grid DOA estimation is studied in the last part of this dissertation. In the first part of this thesis, we devise a sparse array design algorithm for adaptive beamforming. Our strategy is based on finding a sparse beamformer weight to maximize the output signal-to-interference-plus-noise ratio (SINR). The proposed method utilizes ADMM, and admits closed-form solutions at each ADMM iteration. The algorithm convergence properties are analyzed by showing the monotonicity and boundedness of the augmented Lagrangian function. In addition, ...

Data acquisition is a necessary first step in digital signal processing applications such as radar, wireless communications and array processing. Traditionally, this process is performed by uniformly sampling signals at a frequency above the Nyquist rate and converting the resulting samples into digital numeric values through high-resolution amplitude quantization. While the traditional approach to data acquisition is straightforward and extremely well-proven, it may be either impractical or impossible in many modern applications due to the existing fundamental trade-off between sampling rate, amplitude quantization precision, implementation costs, and usage of physical resources, e.g. bandwidth and power consumption. Motivated by this fact, system designers have recently proposed exploiting sparse and few-bit quantized sampling instead of the traditional way of data acquisition in order to reduce implementation costs and usage of physical resources in such applications. However, before transition from the tradition data ...

In direction finding (DF) applications, there are several factors affecting the estimation accuracy of the direction-of-arrivals (DOA) of unknown source locations. The major distortions in the estimation process are due to the array imperfections, model mismatches and multipath. The array imperfections usually exist in practical applications due to the nonidealities in the antenna array such as mutual coupling (MC) and gain/phase uncertainties. The model mismatches usually occur when the model of the received signal differs from the signal model used in the processing stage of the DF system. Another distortion is due to multipath signals. In the multipath scenario, the antenna array receives the transmitted signal from more than one path with different directions and the array covariance matrix is rank-deficient. In this thesis, three new methods are proposed for the problems in DF applications in the presence of array ...

Hyperspectral (HS) imaging, which consists of acquiring a same scene in several hundreds of contiguous spectral bands (a three dimensional data cube), has opened a new range of relevant applications, such as target detection [MS02], classification [C.-03] and spectral unmixing [BDPD+12]. However, while HS sensors provide abundant spectral information, their spatial resolution is generally more limited. Thus, fusing the HS image with other highly resolved images of the same scene, such as multispectral (MS) or panchromatic (PAN) images is an interesting problem. The problem of fusing a high spectral and low spatial resolution image with an auxiliary image of higher spatial but lower spectral resolution, also known as multi-resolution image fusion, has been explored for many years [AMV+11]. From an application point of view, this problem is also important as motivated by recent national programs, e.g., the Japanese next-generation space-borne ...

This dissertation is the combination of three manuscripts –either published in or submitted to journals– on compressive sensing of propeller noise for detection, identification and localization of water crafts. Propeller noise, as a result of rotating blades, is broadband and radiates through water dominating underwater acoustic noise spectrum especially when cavitation develops. Propeller cavitation yields cyclostationary noise which can be modeled by amplitude modulation, i.e., the envelope-carrier product. The envelope consists of the so-called propeller tonals representing propeller characteristics which is used to identify water crafts whereas the carrier is a stationary broadband process. Sampling for propeller noise processing yields large data sizes due to Nyquist rate and multiple sensor deployment. A compressive sensing scheme is proposed for efficient sampling of second-order cyclostationary propeller noise since the spectral correlation function of the amplitude modulation model is sparse as shown in ...

Multi-input and multi-output (MIMO) radars achieve high resolution of arrival direction by transmitting orthogonal waveforms, performing matched filtering at the receiver end and then jointly processing the measurements of all receive antennas. This dissertation studies the use of compressive sensing (CS) and matrix completion (MC) techniques as means of reducing the amount of data that need to be collected by a MIMO radar system, without sacrificing the system’s good resolution properties. MIMO radars with sparse sensing are useful in networked radar scenarios, in which the joint processing of the measurements is done at a fusion center, which might be connected to the receive antennas via a wireless link. In such scenarios, reduced amount of data translates into bandwidth and power saving in the receiver-fusion center link. First, we consider previously defined CS-based MIMO radar schemes, and propose optimal transmit antenna ...

In this thesis we investigate centralized and distributed variants of sparse Bayesian learning (SBL), an effective probabilistic regression method used in machine learning. Since inference in an SBL model is not tractable in closed form, approximations are needed. We focus on the variational Bayesian approximation, as opposed to others used in the literature, for three reasons: First, it is a flexible general framework for approximate Bayesian inference that estimates probability densities including point estimates as a special case. Second, it has guaranteed convergence properties. And third, it is a deterministic approximation concept that is even applicable for high dimensional problems where non-deterministic sampling methods may be prohibitive. We resolve some inconsistencies in the literature involved in other SBL approximation techniques with regard to a proper Bayesian treatment and the incorporation of a very desired property, namely scale invariance. More specifically, ...

In order to find more sophisticated trends in data, potential correlations between larger and larger groups of variables must be considered. Unfortunately, the number of such correlations generally increases exponentially with the number of input variables and, as a result, brute force approaches become unfeasible. So, the data needs to be simplified sufficiently. Yet, the data may not be oversimplified. A method that is widely used for this purpose is to first cast the data as a matrix and the compute a low rank matrix approximation. The tight equivalences between the Weighted Low Rank Approximation (WLRA) problem and the Total Least Squares (TLS) problem are explored. Despite the seemingly different problem formulations of WLRA and TLS, it is shown that both methods can be reduced to the same mathematical kernel problem, i.e. finding the closest (in a certain sense) weighted ...