A System Centric View of Modern Structured and Sparse Inference Tasks

University of Minnesota Ph.D. dissertation.June 2017. Major: Electrical/Computer Engineering. Advisor: Jarvis Haupt. 1 computer file (PDF); xii, 140 pages.We are living in the era of data deluge wherein we are collecting unprecedented amount of data from variety of sources. Modern inference tasks are centered around exploiting structure and sparsity in the data to extract relevant information. This thesis takes an end-to-end system centric view of these inference tasks which mainly consist of two sub-parts (i) data acquisition and (ii) data processing. In context of the data acquisition part of the system, we address issues pertaining to noise, clutter (the unwanted extraneous signals which accompany the desired signal), quantization, and missing observations. In the data processing part of the system we investigate the problems that arise in resource-constrained scenarios such as limited computational power and limited battery life. The first part of this thesis is centered around computationally-efficient approximations of a given linear dimensionality reduction (LDR) operator. In particular, we explore the partial circulant matrix (a matrix whose rows are related by circular shifts) based approximations as they allow for computationally-efficient implementations. We present several theoretical results that provide insight into existence of such approximations. We also propose a data-driven approach to numerically obtain such approximations and demonstrate the utility on real-life data. The second part of this thesis is focused around the issues of noise, missing observations, and quantization arising in matrix and tensor data. In particular, we propose a sparsity regularized maximum likelihood approach to completion of matrices following sparse factor models (matrices which can be expressed as a product of two matrices one of which is sparse). We provide general theoretical error bounds for the proposed approach which can be instantiated for variety of noise distributions. We also consider the problem of tensor completion and extend the results of matrix completion to the tensor setting. The problem of matrix completion from quantized and noisy observations is also investigated in as general terms as possible. We propose a constrained maximum likelihood approach to quantized matrix completion, provide probabilistic error bounds for this approach, and numerical algorithms which are used to provide numerical evidence for the proposed error bounds. The final part of this thesis is focused on issues related to clutter and limited battery life in signal acquisition. Specifically, we investigate the problem of compressive measurement design under a given sensing energy budget for estimating structured signals in structured clutter. We propose a novel approach that leverages the prior information about signal and clutter to judiciously allocate sensing energy to the compressive measurements. We also investigate the problem of processing Electrodermal Activity (EDA) signals recorded as the conductance over a user's skin. EDA signals contain information about the user's neuron ring and psychological state. These signals contain the desired information carrying signal superimposed with unwanted components which may be considered as clutter. We propose a novel compressed sensing based approach with provable error guarantees for processing EDA signals to extract relevant information, and demonstrate its efficacy, as compared to existing techniques, via numerical experiments