Localization is a fundamental operation for many wireless networks. While GPS is widely used for location determination, it is unavailable in many environments either due to its high cost or the lack of line of sight to the satellites (e.g., indoors, under the ground, or in a downtown canyon). The limitations of GPS have motivated researchers to develop many localization schemes to infer locations based on measured wireless signals. However, most of these existing schemes focus on localization in static wireless networks. As many wireless networks are mobile (e.g., mobile sensor networks, disaster recovery networks, and vehicular networks), we focus on localization in mobile networks in this thesis. We analyze real mobility traces and find that they exhibit temporal stability and low-rank structure. Motivated by this observation, we develop three novel localization schemes to accurately determine locations in mobile networks: 1. Low Rank based Localization (LRL), which exploits the low-rank structure in mobility. 2. Temporal Stability based Localization (TSL), which leverages the temporal stability. 3. Temporal Stability and Low Rank based Localization (TSLRL), which incorporates both the temporal stability and the low-rank structure. These localization schemes are general and can leverage either mere connectivity (i.e., range-free localization) or distance estimation between neighbors (i.e., range-based localization). Using extensive simulations and testbed experiments, we show that our new schemes significantly outperform state-of-the-art localization schemes under a wide range of scenarios and are robust to measurement errors. / text

If a structure is composed mainly of one material but contains a small amount of a second material, and if these two materials have significantly different levels of structural damping, this can increase the cost of solving the modal frequency response problem substantially. Even if the rank of the contribution to the finite element structural damping matrix from the second material is very low, the matrix becomes fully populated when transformed to the modal representation. As a result, the complex-valued modal matrix that represents the structure’s stiffness and structural damping is both full rank, because of the diagonal part contributed by the stiffness, and fully populated, because of off-diagonal imaginary terms contributed by the second material’s structural damping. Solving the modal frequency response problem at many frequencies requires either the factorization of a coefficient matrix at every frequency, or the solution of a complex symmetric eigenvalue problem associated with the modal stiffness/structural damping matrix. The cost of both of these approaches is proportional to the cube of the number of modes included in the analysis. This cost could be reduced greatly if the damping properties of the structure were handled carefully in modeling the structure, but in practical computation of the modal frequency response, the information that could potentially reduce the computational cost is often unavailable. This thesis explores the possibilities of obtaining a representation of the complex modal stiffness/structural damping matrix as a diagonal matrix plus a matrix of minimal rank. An algorithm for computing a “diagonal plus low rank” (DPLR) representation is developed, along with an iterative algorithm for using an inexact DPLR approximation in the solution of the modal frequency response problem. The behavior of these algorithms is investigated on several example problems. / text

We discuss the numerical solution of large-scale sparse projected discrete-time periodic Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

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Low-dimensional signal structures naturally arise in a large set of applications in various fields such as medical imaging, machine learning, signal, and array processing. A ubiquitous low-dimensional structure in signals and images is sparsity, and a new sampling theory; namely, compressive sensing, proves that the sparse signals and images can be reconstructed from incomplete measurements. The signal recovery is achieved using efficient algorithms such as \ell_1-minimization. Recently, the research focus has spun-off to encompass other interesting low-dimensional signal structures such as group-sparsity and low-rank structure. This thesis considers low-rank matrix recovery (LRMR) from various structured-random measurement ensembles. These results are then employed for the in depth investigation of the classical blind-deconvolution problem from a new perspective, and for the development of a framework for the efficient sampling of correlated signals (the signals lying in a subspace). In the first part, we study the blind deconvolution; separation of two unknown signals by observing their convolution. We recast the deconvolution of discrete signals w and x as a rank-1 matrix wx* recovery problem from a structured random measurement ensemble. The convex relaxation of the problem leads to a tractable semidefinite program. We show, using some of the mathematical tools developed recently for LRMR, that if we assume the signals convolved with one another live in known subspaces, then this semidefinite relaxation is provably effective. In the second part, we design various efficient sampling architectures for signals acquired using large arrays. The sampling architectures exploit the correlation in the signals to acquire them at a sub-Nyquist rate. The sampling devices are designed using analog components with clear implementation potential. For each of the sampling scheme, we show that the signal reconstruction can be framed as an LRMR problem from a structured-random measurement ensemble. The signals can be reconstructed using the familiar nuclear-norm minimization. The sampling theorems derived for each of the sampling architecture show that the LRMR framework produces the Shannon-Nyquist performance for the sub-Nyquist acquisition of correlated signals. In the final part, we study low-rank matrix factorizations using randomized linear algebra. This specific method allows us to use a least-squares program for the reconstruction of the unknown low-rank matrix from the samples of its row and column space. Based on the principles of this method, we then design sampling architectures that not only acquire correlated signals efficiently but also require a simple least-squares program for the signal reconstruction. A theoretical analysis of all of the LRMR problems above is presented in this thesis, which provides the sufficient measurements required for the successful reconstruction of the unknown low-rank matrix, and the upper bound on the recovery error in both noiseless and noisy cases. For each of the LRMR problem, we also provide a discussion of a computationally feasible algorithm, which includes a least-squares-based algorithm, and some of the fastest algorithms for solving nuclear-norm minimization.

In many machine learning problems and application domains, the data are naturally organized by groups. For example, a video sequence is a group of images, an image is a group of patches, a document is a group of paragraphs/words, and a community is a group of people. We call them the collective data. In this thesis, we study how and what we can learn from collective data. Usually, machine learning focuses on individual objects, each of which is described by a feature vector and studied as a point in some metric space. When approaching collective data, researchers often reduce the groups into vectors to which traditional methods can be applied. We, on the other hand, will try to develop machine learning methods that respect the collective nature of data and learn from them directly. Several different approaches were taken to address this learning problem. When the groups consist of unordered discrete data points, it can naturally be characterized by its sufficient statistics – the histogram. For this case we develop efficient methods to address the outliers and temporal effects in the data based on matrix and tensor factorization methods. To learn from groups that contain multi-dimensional real-valued vectors, we develop both generative methods based on hierarchical probabilistic models and discriminative methods using group kernels based on new divergence estimators. With these tools, we can accomplish various tasks such as classification, regression, clustering, anomaly detection, and dimensionality reduction on collective data. We further consider the practical side of the divergence based algorithms. To reduce their time and space requirements, we evaluate and find methods that can effectively reduce the size of the groups with little impact on the accuracy. We also proposed the conditional divergence along with an efficient estimator in order to correct the sampling biases that might be present in the data. Finally, we develop methods to learn in cases where some divergences are missing, caused by either insufficient computational resources or extreme sampling biases. In addition to designing new learning methods, we will use them to help the scientific discovery process. In our collaboration with astronomers and physicists, we see that the new techniques can indeed help scientists make the best of data.

COOPERATIVE WIDEBAND SPECTRUM SENSING BASED ON JOINT SPARSITY By Ghazaleh Jowkar, Master of Science A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University Virginia Commonwealth University 2017 Major Director: Dr. Ruixin Niu, Associate Professor of Department of Electrical and Computer Engineering In this thesis, the problem of wideband spectrum sensing in cognitive radio (CR) networks using sub-Nyquist sampling and sparse signal processing techniques is investigated. To mitigate multi-path fading, it is assumed that a group of spatially dispersed SUs collaborate for wideband spectrum sensing, to determine whether or not a channel is occupied by a primary user (PU). Due to the underutilization of the spectrum by the PUs, the spectrum matrix has only a small number of non-zero rows. In existing state-of-the-art approaches, the spectrum sensing problem was solved using the low-rank matrix completion technique involving matrix nuclear-norm minimization. Motivated by the fact that the spectrum matrix is not only low-rank, but also sparse, a spectrum sensing approach is proposed based on minimizing a mixed-norm of the spectrum matrix instead of low-rank matrix completion to promote the joint sparsity among the column vectors of the spectrum matrix. Simulation results are obtained, which demonstrate that the proposed mixed-norm minimization approach outperforms the low-rank matrix completion based approach, in terms of the PU detection performance. Further we used mixed-norm minimization model in multi time frame detection. Simulation results shows that increasing the number of time frames will increase the detection performance, however, by increasing the number of time frames after a number of times the performance decrease dramatically.

Quantum-mechanical effects play a major role in the performance of modern electronic devices. In order to predict the behavior of novel devices, quantum effects are often included using Non-Equilibrium Green's Function (NEGF) methods in atomistic device representations. These quantum effects may include realistic inelastic scattering caused by device impurities and phonons. With the inclusion of realistic physical phenomena, the computational load of predictive simulations increases greatly, and a manageable basis through low-rank approximations is desired.<br><br>In this work, low-rank approximations are used to reduce the computational load of atomistic simulations. The benefits of basis reductions on simulation time and peak memory are assessed.<br>The low-rank approximation method is then extended to include more realistic physical effects than those modeled today, including exact calculations of scattering phenomena. The inclusion of these exact calculations are then contrasted to current methods and approximations.

<p>Many kernel matrices from differential equations or data science applications possess low or approximately low off-diagonal rank for certain key matrix subblocks; such matrices are referred to as rank-structured. Operations on rank-structured matrices like factorization and linear system solution can be greatly accelerated by converting them into hierarchical matrix forms, such as the hiearchically semiseparable (HSS) matrix form. The dominant cost of this conversion process, called HSS construction, is the low-rank approximation of certain matrix blocks. Low-rank approximation is also a required step in many other contexts throughout numerical linear algebra. In this work, a proxy point low-rank approximation method is detailed for general analytic kernel matrices, in both one and several dimensions. A new accuracy analysis for this approximation is also provided, as well as numerical evidence of its accuracy. The extension of this method to kernels in several dimensions is novel, and its new accuracy analysis makes it a convenient choice to use over existing proxy point methods. Finally, a new HSS construction algorithm using this method for certain Cauchy and Toeplitz matrices is given, which is asymptotically faster than existing methods. Numerical evidence for the accuracy and efficacy of the new construction algorithm is also provided.</p>

Indiana University-Purdue University Indianapolis (IUPUI) / The successful treatment and potential eradication of many complex diseases, such as cancer, begins with elucidating the convoluted mapping of molecular profiles to phenotypical manifestation. Our observed molecular profiles (e.g., genomics, transcriptomics, epigenomics) are often high-dimensional and are collected from patient samples falling into heterogeneous disease subtypes. Interpretable learning from such data calls for sparsity-driven models. This dissertation addresses the high dimensionality, sparsity, and heterogeneity issues when analyzing multiple-omics data, where each method is implemented with a concomitant R package. First, we examine challenges in submatrix identification, which aims to find subgroups of samples that behave similarly across a subset of features. We resolve issues such as two-way sparsity, non-orthogonality, and parameter tuning with an adaptive thresholding procedure on the singular vectors computed via orthogonal iteration. We validate the method with simulation analysis and apply it to an Alzheimer’s disease dataset. The second project focuses on modeling relationships between large, matched datasets. Exploring regressional structures between large data sets can provide insights such as the effect of long-range epigenetic influences on gene expression. We present a high-dimensional version of mixture multivariate regression to detect patient clusters, each with different correlation structures of matched-omics datasets. Results are validated via simulation and applied to matched-omics data sets. In the third project, we introduce a novel approach to modeling spatial transcriptomics (ST) data with a spatially penalized multinomial model of the expression counts. This method solves the low-rank structures of zero-inflated ST data with spatial smoothness constraints. We validate the model using manual cell structure annotations of human brain samples. We then applied this technique to additional ST datasets. / 2025-05-22