Scalable Low-rank Matrix and Tensor Decomposition on Graphs

In many signal processing, machine learning and computer vision applications, one often has to deal with high dimensional and big datasets such as images, videos, web content, etc. The data can come in various forms, such as univariate or multivariate time series, matrices or high dimensional tensors. The goal of the data mining community is to reveal the hidden linear or non-linear structures in the datasets. Over the past couple of decades matrix factorization, owing to its intrinsic association with dimensionality reduction has been adopted as one of the key methods in this context. One can either use a single linear subspace to approximate the data (the standard Principal Component Analysis (PCA) approach) or a union of low dimensional subspaces where each data class belongs to a different subspace. In many cases, however, the low dimensional data follows some additional structure. Knowledge of such structure is beneficial, as we can use it to enhance the representativity of our models by adding structured priors. A nowadays standard way to represent pairwise affinity between objects is by using graphs. The introduction of graph-based priors to enhance matrix factorization models has recently brought them back to the highest attention of the data mining community. Representation of a signal on a graph is well motivated by the emerging field of signal processing on graphs, based on notions of spectral graph theory. The underlying assumption is that high-dimensional data samples lie on or close to a smooth low-dimensional manifold. Interestingly, the underlying manifold can be represented by its discrete proxy, i.e. a graph. A primary limitation of the state-of-the-art low-rank approximation methods is that they do not generalize for the case of non-linear low-rank structures. Furthermore, the standard low-rank extraction methods for many applications, such as low-rank and sparse decomposition, are computationally cumbersome. We argue, that for many machine learning and signal processing applications involving big data, an approximate low-rank recovery suffices. Thus, in this thesis, we present solutions to the above two limitations by presenting a new framework for scalable but approximate low-rank extraction which exploits the hidden structure in the data using the notion of graphs. First, we present a novel signal model, called `Multilinear low-rank tensors on graphs (MLRTG)' which states that a tensor can be encoded as a multilinear combination of the low-frequency graph eigenvectors, where the graphs are constructed along the various modes of the tensor. Since the graph eigenvectors have the interpretation of \textit{non-linear} embedding of a dataset on the low-dimensional manifold, we propose a method called `Graph Multilinear SVD (GMLSVD)' to recover PCA based linear subspaces from these eigenvectors. Finally, we propose a plethora of highly scalable matrix and tensor based problems for low-rank extraction which implicitly or explicitly make use of the GMLSVD framework. The core idea is to replace the expensive iterative SVD operations by updating the linear subspaces from the fixed non-linear ones via low-cost operations. We present applications in low-rank and sparse decomposition and clustering of the low-rank features to evaluate all the proposed methods. Our theoretical analysis shows that the approximation error of the proposed framework depends on the spectral properties of the graph Laplacian

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

Low-rank and sparse recovery of human gait data

Due to occlusion or detached markers, information can often be lost while capturing human motion with optical tracking systems. Based on three natural properties of human gait movement, this study presents two different approaches to recover corrupted motion data. These properties are used to define a reconstruction model combining low-rank matrix completion of the measured data with a group-sparsity prior on the marker trajectories mapped in the frequency domain. Unlike most existing approaches, the proposed methodology is fully unsupervised and does not need training data or kinematic information of the user. We evaluated our methods on four different gait datasets with various gap lengths and compared their performance with a state-of-the-art approach using principal component analysis (PCA). Our results showed recovering missing data more precisely, with a reduction of at least 2 mm in mean reconstruction error compared to the literature method. When a small number of marker trajectories is available, our findings showed a reduction of more than 14 mm for the mean reconstruction error compared to the literature approach

STRUCTURED SPARSITY DRIVEN LEARNING: THEORY AND ALGORITHMS

Ph.DDOCTOR OF PHILOSOPH

Compressive Acquisition and Processing of Sparse Analog Signals

Since the advent of the first digital processing units, the importance of digital signal processing has been steadily rising. Today, most signal processing happens in the digital domain, requiring that analog signals be first sampled and digitized before any relevant data can be extracted from them. The recent explosion of the demands for data acquisition, storage and processing, however, has pushed the capabilities of conventional acquisition systems to their limits in many application areas. By offering an alternative view on the signal acquisition process, ideas from sparse signal processing and one of its main beneficiaries compressed sensing (CS), aim at alleviating some of these problems. In this thesis, we look into the ways the application of a compressive measurement kernel impacts the signal recovery performance and investigate methods to infer the current signal complexity from the compressive observations. We then study a particular application, namely that of sub-Nyquist sampling and processing of sparse analog multiband signals in spectral, angular and spatial domains.Seit dem Aufkommen der ersten digitalen Verarbeitungseinheiten hat die Bedeutung der digitalen Signalverarbeitung stetig zugenommen. Heutzutage findet die meiste Signalverarbeitung im digitalen Bereich statt, was erfordert, dass analoge Signale zuerst abgetastet und digitalisiert werden, bevor relevante Daten daraus extrahiert werden können. Jahrzehntelang hat die herkömmliche äquidistante Abtastung, die durch das Nyquist-Abtasttheorem bestimmt wird, zu diesem Zweck ein nahezu universelles Mittel bereitgestellt. Der kürzliche explosive Anstieg der Anforderungen an die Datenerfassung, -speicherung und -verarbeitung hat jedoch die Fähigkeiten herkömmlicher Erfassungssysteme in vielen Anwendungsbereichen an ihre Grenzen gebracht. Durch eine alternative Sichtweise auf den Signalerfassungsprozess können Ideen aus der sparse Signalverarbeitung und einer ihrer Hauptanwendungsgebiete, Compressed Sensing (CS), dazu beitragen, einige dieser Probleme zu mindern. Basierend auf der Annahme, dass der Informationsgehalt eines Signals oft viel geringer ist als was von der nativen Repräsentation vorgegeben, stellt CS ein alternatives Konzept für die Erfassung und Verarbeitung bereit, das versucht, die Abtastrate unter Beibehaltung des Signalinformationsgehalts zu reduzieren. In dieser Arbeit untersuchen wir einige der Grundlagen des endlichdimensionalen CSFrameworks und seine Verbindung mit Sub-Nyquist Abtastung und Verarbeitung von sparsen analogen Signalen. Obwohl es seit mehr als einem Jahrzehnt ein Schwerpunkt aktiver Forschung ist, gibt es noch erhebliche Lücken beim Verständnis der Auswirkungen von komprimierenden Ansätzen auf die Signalwiedergewinnung und die Verarbeitungsleistung, insbesondere bei rauschbehafteten Umgebungen und in Bezug auf praktische Messaufgaben. In dieser Dissertation untersuchen wir, wie sich die Anwendung eines komprimierenden Messkerns auf die Signal- und Rauschcharakteristiken auf die Signalrückgewinnungsleistung auswirkt. Wir erforschen auch Methoden, um die aktuelle Signal-Sparsity-Order aus den komprimierten Messungen abzuleiten, ohne auf die Nyquist-Raten-Verarbeitung zurückzugreifen, und zeigen den Vorteil, den sie für den Wiederherstellungsprozess bietet. Nachdem gehen wir zu einer speziellen Anwendung, nämlich der Sub-Nyquist-Abtastung und Verarbeitung von sparsen analogen Multibandsignalen. Innerhalb des Sub-Nyquist-Abtastung untersuchen wir drei verschiedene Multiband-Szenarien, die Multiband-Sensing in der spektralen, Winkel und räumlichen-Domäne einbeziehen.Since the advent of the first digital processing units, the importance of digital signal processing has been steadily rising. Today, most signal processing happens in the digital domain, requiring that analog signals be first sampled and digitized before any relevant data can be extracted from them. For decades, conventional uniform sampling that is governed by the Nyquist sampling theorem has provided an almost universal means to this end. The recent explosion of the demands for data acquisition, storage and processing, however, has pushed the capabilities of conventional acquisition systems to their limits in many application areas. By offering an alternative view on the signal acquisition process, ideas from sparse signal processing and one of its main beneficiaries compressed sensing (CS), have the potential to assist alleviating some of these problems. Building on the premise that the signal information rate is often much lower than what is dictated by its native representation, CS provides an alternative acquisition and processing framework that attempts to reduce the sampling rate while preserving the information content of the signal. In this thesis, we explore some of the basic foundations of the finite-dimensional CS framework and its connection to sub-Nyquist sampling and processing of sparse continuous analog signals with application to multiband sensing. Despite being a focus of active research for over a decade, there still remain signi_cant gaps in understanding the implications that compressive approaches have on the signal recovery and processing performance, especially against noisy settings and in relation to practical sampling problems. This dissertation aims at filling some of these gaps. More specifically, we look into the ways the application of a compressive measurement kernel impacts signal and noise characteristics and the relation it has to the signal recovery performance. We also investigate methods to infer the current complexity of the signal scene from the reduced-rate compressive observations without resorting to Nyquist-rate processing and show the advantage this knowledge offers to the recovery process. Having considered some of the universal aspects of compressive systems, we then move to studying a particular application, namely that of sub-Nyquist sampling and processing of sparse analog multiband signals. Within the sub-Nyquist sampling framework, we examine three different multiband scenarios that involve multiband sensing in spectral, angular and spatial domains. For each of them, we provide a sub-Nyquist receiver architecture, develop recovery methods and numerically evaluate their performance

Remote Sensing Data Compression

A huge amount of data is acquired nowadays by different remote sensing systems installed on satellites, aircrafts, and UAV. The acquired data then have to be transferred to image processing centres, stored and/or delivered to customers. In restricted scenarios, data compression is strongly desired or necessary. A wide diversity of coding methods can be used, depending on the requirements and their priority. In addition, the types and properties of images differ a lot, thus, practical implementation aspects have to be taken into account. The Special Issue paper collection taken as basis of this book touches on all of the aforementioned items to some degree, giving the reader an opportunity to learn about recent developments and research directions in the field of image compression. In particular, lossless and near-lossless compression of multi- and hyperspectral images still remains current, since such images constitute data arrays that are of extremely large size with rich information that can be retrieved from them for various applications. Another important aspect is the impact of lossless compression on image classification and segmentation, where a reasonable compromise between the characteristics of compression and the final tasks of data processing has to be achieved. The problems of data transition from UAV-based acquisition platforms, as well as the use of FPGA and neural networks, have become very important. Finally, attempts to apply compressive sensing approaches in remote sensing image processing with positive outcomes are observed. We hope that readers will find our book useful and interestin