Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis

The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift towards models that are essentially polynomial and whose uniqueness, unlike the matrix methods, is guaranteed under verymild and natural conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints that match data properties, and to find more general latent components in the data than matrix-based methods. A comprehensive introduction to tensor decompositions is provided from a signal processing perspective, starting from the algebraic foundations, via basic Canonical Polyadic and Tucker models, through to advanced cause-effect and multi-view data analysis schemes. We show that tensor decompositions enable natural generalizations of some commonly used signal processing paradigms, such as canonical correlation and subspace techniques, signal separation, linear regression, feature extraction and classification. We also cover computational aspects, and point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated with big datasets. The concepts are supported by illustrative real world case studies illuminating the benefits of the tensor framework, as efficient and promising tools for modern signal processing, data analysis and machine learning applications; these benefits also extend to vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker decomposition, HOSVD, tensor networks, Tensor Train

Advanced Algebraic Concepts for Efficient Multi-Channel Signal Processing

Unsere moderne Gesellschaft ist Zeuge eines fundamentalen Wandels in der Art und Weise wie wir mit Technologie interagieren. Geräte werden zunehmend intelligenter - sie verfügen über mehr und mehr Rechenleistung und häufiger über eigene Kommunikationsschnittstellen. Das beginnt bei einfachen Haushaltsgeräten und reicht über Transportmittel bis zu großen überregionalen Systemen wie etwa dem Stromnetz. Die Erfassung, die Verarbeitung und der Austausch digitaler Informationen gewinnt daher immer mehr an Bedeutung. Die Tatsache, dass ein wachsender Anteil der Geräte heutzutage mobil und deshalb batteriebetrieben ist, begründet den Anspruch, digitale Signalverarbeitungsalgorithmen besonders effizient zu gestalten. Dies kommt auch dem Wunsch nach einer Echtzeitverarbeitung der großen anfallenden Datenmengen zugute. Die vorliegende Arbeit demonstriert Methoden zum Finden effizienter algebraischer Lösungen für eine Vielzahl von Anwendungen mehrkanaliger digitaler Signalverarbeitung. Solche Ansätze liefern nicht immer unbedingt die bestmögliche Lösung, kommen dieser jedoch häufig recht nahe und sind gleichzeitig bedeutend einfacher zu beschreiben und umzusetzen. Die einfache Beschreibungsform ermöglicht eine tiefgehende Analyse ihrer Leistungsfähigkeit, was für den Entwurf eines robusten und zuverlässigen Systems unabdingbar ist. Die Tatsache, dass sie nur gebräuchliche algebraische Hilfsmittel benötigen, erlaubt ihre direkte und zügige Umsetzung und den Test unter realen Bedingungen. Diese Grundidee wird anhand von drei verschiedenen Anwendungsgebieten demonstriert. Zunächst wird ein semi-algebraisches Framework zur Berechnung der kanonisch polyadischen (CP) Zerlegung mehrdimensionaler Signale vorgestellt. Dabei handelt es sich um ein sehr grundlegendes Werkzeug der multilinearen Algebra mit einem breiten Anwendungsspektrum von Mobilkommunikation über Chemie bis zur Bildverarbeitung. Verglichen mit existierenden iterativen Lösungsverfahren bietet das neue Framework die Möglichkeit, den Rechenaufwand und damit die Güte der erzielten Lösung zu steuern. Es ist außerdem weniger anfällig gegen eine schlechte Konditionierung der Ausgangsdaten. Das zweite Gebiet, das in der Arbeit besprochen wird, ist die unterraumbasierte hochauflösende Parameterschätzung für mehrdimensionale Signale, mit Anwendungsgebieten im RADAR, der Modellierung von Wellenausbreitung, oder bildgebenden Verfahren in der Medizin. Es wird gezeigt, dass sich derartige mehrdimensionale Signale mit Tensoren darstellen lassen. Dies erlaubt eine natürlichere Beschreibung und eine bessere Ausnutzung ihrer Struktur als das mit Matrizen möglich ist. Basierend auf dieser Idee entwickeln wir eine tensor-basierte Schätzung des Signalraums, welche genutzt werden kann um beliebige existierende Matrix-basierte Verfahren zu verbessern. Dies wird im Anschluss exemplarisch am Beispiel der ESPRIT-artigen Verfahren gezeigt, für die verbesserte Versionen vorgeschlagen werden, die die mehrdimensionale Struktur der Daten (Tensor-ESPRIT), nichzirkuläre Quellsymbole (NC ESPRIT), sowie beides gleichzeitig (NC Tensor-ESPRIT) ausnutzen. Um die endgültige Schätzgenauigkeit objektiv einschätzen zu können wird dann ein Framework für die analytische Beschreibung der Leistungsfähigkeit beliebiger ESPRIT-artiger Algorithmen diskutiert. Verglichen mit existierenden analytischen Ausdrücken ist unser Ansatz allgemeiner, da keine Annahmen über die statistische Verteilung von Nutzsignal und Rauschen benötigt werden und die Anzahl der zur Verfügung stehenden Schnappschüsse beliebig klein sein kann. Dies führt auf vereinfachte Ausdrücke für den mittleren quadratischen Schätzfehler, die Schlussfolgerungen über die Effizienz der Verfahren unter verschiedenen Bedingungen zulassen. Das dritte Anwendungsgebiet ist der bidirektionale Datenaustausch mit Hilfe von Relay-Stationen. Insbesondere liegt hier der Fokus auf Zwei-Wege-Relaying mit Hilfe von Amplify-and-Forward-Relays mit mehreren Antennen, da dieser Ansatz ein besonders gutes Kosten-Nutzen-Verhältnis verspricht. Es wird gezeigt, dass sich die nötige Kanalkenntnis mit einem einfachen algebraischen Tensor-basierten Schätzverfahren gewinnen lässt. Außerdem werden Verfahren zum Finden einer günstigen Relay-Verstärkungs-Strategie diskutiert. Bestehende Ansätze basieren entweder auf komplexen numerischen Optimierungsverfahren oder auf Ad-Hoc-Ansätzen die keine zufriedenstellende Bitfehlerrate oder Summenrate liefern. Deshalb schlagen wir algebraische Ansätze zum Finden der Relayverstärkungsmatrix vor, die von relevanten Systemmetriken inspiriert sind und doch einfach zu berechnen sind. Wir zeigen das algebraische ANOMAX-Verfahren zum Erreichen einer niedrigen Bitfehlerrate und seine Modifikation RR-ANOMAX zum Erreichen einer hohen Summenrate. Für den Spezialfall, in dem die Endgeräte nur eine Antenne verwenden, leiten wir eine semi-algebraische Lösung zum Finden der Summenraten-optimalen Strategie (RAGES) her. Anhand von numerischen Simulationen wird die Leistungsfähigkeit dieser Verfahren bezüglich Bitfehlerrate und erreichbarer Datenrate bewertet und ihre Effektivität gezeigt.Modern society is undergoing a fundamental change in the way we interact with technology. More and more devices are becoming "smart" by gaining advanced computation capabilities and communication interfaces, from household appliances over transportation systems to large-scale networks like the power grid. Recording, processing, and exchanging digital information is thus becoming increasingly important. As a growing share of devices is nowadays mobile and hence battery-powered, a particular interest in efficient digital signal processing techniques emerges. This thesis contributes to this goal by demonstrating methods for finding efficient algebraic solutions to various applications of multi-channel digital signal processing. These may not always result in the best possible system performance. However, they often come close while being significantly simpler to describe and to implement. The simpler description facilitates a thorough analysis of their performance which is crucial to design robust and reliable systems. The fact that they rely on standard algebraic methods only allows their rapid implementation and test under real-world conditions. We demonstrate this concept in three different application areas. First, we present a semi-algebraic framework to compute the Canonical Polyadic (CP) decompositions of multidimensional signals, a very fundamental tool in multilinear algebra with applications ranging from chemistry over communications to image compression. Compared to state-of-the art iterative solutions, our framework offers a flexible control of the complexity-accuracy trade-off and is less sensitive to badly conditioned data. The second application area is multidimensional subspace-based high-resolution parameter estimation with applications in RADAR, wave propagation modeling, or biomedical imaging. We demonstrate that multidimensional signals can be represented by tensors, providing a convenient description and allowing to exploit the multidimensional structure in a better way than using matrices only. Based on this idea, we introduce the tensor-based subspace estimate which can be applied to enhance existing matrix-based parameter estimation schemes significantly. We demonstrate the enhancements by choosing the family of ESPRIT-type algorithms as an example and introducing enhanced versions that exploit the multidimensional structure (Tensor-ESPRIT), non-circular source amplitudes (NC ESPRIT), and both jointly (NC Tensor-ESPRIT). To objectively judge the resulting estimation accuracy, we derive a framework for the analytical performance assessment of arbitrary ESPRIT-type algorithms by virtue of an asymptotical first order perturbation expansion. Our results are more general than existing analytical results since we do not need any assumptions about the distribution of the desired signal and the noise and we do not require the number of samples to be large. At the end, we obtain simplified expressions for the mean square estimation error that provide insights into efficiency of the methods under various conditions. The third application area is bidirectional relay-assisted communications. Due to its particularly low complexity and its efficient use of the radio resources we choose two-way relaying with a MIMO amplify and forward relay. We demonstrate that the required channel knowledge can be obtained by a simple algebraic tensor-based channel estimation scheme. We also discuss the design of the relay amplification matrix in such a setting. Existing approaches are either based on complicated numerical optimization procedures or on ad-hoc solutions that to not perform well in terms of the bit error rate or the sum-rate. Therefore, we propose algebraic solutions that are inspired by these performance metrics and therefore perform well while being easy to compute. For the MIMO case, we introduce the algebraic norm maximizing (ANOMAX) scheme, which achieves a very low bit error rate, and its extension Rank-Restored ANOMAX (RR-ANOMAX) that achieves a sum-rate close to an upper bound. Moreover, for the special case of single antenna terminals we derive the semi-algebraic RAGES scheme which finds the sum-rate optimal relay amplification matrix based on generalized eigenvectors. Numerical simulations evaluate the resulting system performance in terms of bit error rate and system sum rate which demonstrates the effectiveness of the proposed algebraic solutions

Scaling transform based information geometry method for DOA estimation

By exploiting the relationship between probability density and the differential geometry structure of received data and geodesic distance, the recently proposed information geometry (IG) method can provide higher accuracy and resolution ability for direction of arrival (DOA) estimation than many existing methods. However, its performance is not robust even for high signal to noise ratio (SNR). To have a deep understanding of its unstable performance, a theoretical analysis of the IG method is presented by deriving the relationship between the cost function and the number of array elements, powers and DOAs of source signals, and noise power. Then, to make better use of the nonlinear and super resolution property of the cost function, a Scaling TRansform based INformation Geometry (STRING) method is proposed, which simply scales the array received data or its covariance matrix by a real number. However, the expression for the optimum value of the scalar is complicated and related to the unknown signal DOAs and powers. Hence, a decision criterion and a simple search based procedure are developed, guaranteeing a robust performance. As demonstrated by computer simulations, the proposed STRING method has the best and robust angle resolution performance compared with many existing high resolution methods and even outperforms the classic Cramer-Rao bound (CRB), although at the cost of a bias in the estimation results

Time-delay estimation under non-clustered and clustered scenarios for GNSS signals

Tese (doutorado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2021.Aplicações que empregam sistemas globais de navegação por satélite, do inglês Global Navigation Satellite Systems (GNSS) para prover posicionamento acurado estão sujeitos a degradação drástica não só por intereferências eletromagnéticas, como também componentes de multipercurso causados por reflexões e refrações no ambiente. Aplicações de segurança crítica como veículos autonômos e aviação civil, e aplicações de risco crítico como gestão de pesca, pedágio automático, e agricultura de precisão dependem de posicionamento acurado sob cenários complicados. Tipicamente quanto mais agrupamento ocorre entre o componente de linha de visada, do inglês line-of-sight (LOS) e componentes de multipercurso ou não-linha de visada, do inglês non-line-of-sight (NLOS), menos acurada é a estimação da posição. Abordagens tensorials estado da arte para receptores GNSS baseado em arranjos de antenas utilizam processamento tensorial de sinais para separar o componente LOS dos componentes NLOS, assim mitigando os efeitos destes, utilizando decomposição em valores singulares multilinear, do inglês multilinear singular value decomposition (MLSVD) para gerar um autofiltro de order superior, do inglês higher-order eigenfilter (HOE) com pré-processamento por média frente-costas, do inglês forward-backward averaging (FBA), e suavização espacial expandida, do inglês expanded spatial smoothing (ESPS), estimação de direção de chegada, do inglês direction of arrival (DoA) e fatorização Khatri-Rao, do inglês Khatri-Rao factorization (KRF), estimação de Procrustes e fatorização Khatri-Rao (ProKRaft), e o sistema semi-algébrico de decomposição poliádica canônica por diagonalização matricial simultânea, do inglês semi-algebraic framework for approximate canonical polyadic decomposition via simultaneous matrix diagonalization (SECSI), respectivamente. Propomos duas abordagens de processamento para estimação de atraso, do inglês time-delay estimation (TDE). A primeira é a abordagem em lotes utilizando dados de vários períodos do sinal. Usando estimação em lotes propomos duas abordagens algébricas para TDE, em que diagonalizaçao é efetivada por decomposição generalizada em autovalores, do inglês generalized eigenvalue decomposition (GEVD), das primeiras duas fatias frontais do tensor núcleo do tensor de dados, estimado por MLSVD. Esta primeira abordagem, como os métodos citados, na quais simulações foram feitas com 1 componente LOS e 1 componente NLOS, assim os dados observados tem posto cheio em todos seus modos, não faz suposições sobre o posto do tensor de dados. A segunda abordagem supõe cenários nos quais mais de 1 componente NLOS está presente e são agregados (clustered em inglês), assim vários vetores de uma das matrizes-fator que formam o tensor de dados são altamente correlacionaiii dos, resultando num tensor de dados que é de posto deficiente em pelo menos um modo. Os esquemas algébricos baseados em tensores propostos utilizam a decomposição poliádica canônica por decomposição generalizada em autovalores, do inglês canonical polyadic decomposition via generalized eigenvalue decomposition (CPD-GEVD), e a decomposição em termos de posto-(Lr, Lr, 1) por decomposição generalizada em autovalores, do inglês decomposition in multilinear rank-(Lr, Lr, 1) terms via generalized eigenvalue decomposition ((Lr, Lr, 1)-GEVD) para melhorar a TDE do componente LOS sob cenários desafiadores. A segunda é a abordagem de processamento adaptativo de amostras individuais utilizando rastreamento de subespaço a cada período de código, epoch em inglês. Usando processamento adaptativo propomos duas abordagem, uma aplicando FBA expandido (EFBA) e ESPS ao dados e estimando um HOE, e outra usando usa estimação paramétrica para estimar a DoA. Estendendo o modelo para um arranjo retangular uniforme, do inglês uniform rectangular array (URA), o fluxo de dados são tensores de terceira ordem. Para este modelo propomos três abordagens para TDE baseado em HOE, CPD-GEVD, e ESPRIT tensorial, respectivamente e empregando uma estratégia de truncamento sequencial para reduzir a quantidade de operações necessárias para cada modo do tensorCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).Applications employing Global Navigation Satellite Systems (GNSS) to provide accurate positioning are subject to drastic degradation not only due to electromagnetic interference, but also due to multipath components caused by reflections and refractions in the environment. Safety-critical applications such as autonomous vehicles and civil aviation, and liability-critical applications such as fisheries management, automatic tolling, and precision agriculture depend on accurate positioning under such demanding scenarios. Typically, the more clustering occurs between the line-of-sight (LOS) component and multipath or non-line-of-sight (NLOS) components, the more inaccurate is the estimation of the positioning. State-of-the-art tensor based approaches for antenna array-based GNSS receivers apply tensor-based signal processing to separate the LOS components from NLOS components, thus mitigating the effects of the latter, using the multilinear singular value decomposition (MLSVD) to generate a higher-order eigenfilter (HOE) with forward-backward averaging (FBA) and expanded spatial smoothing (ESPS) preprocessing, direction of arrival (DoA) estimation and Khatri-Rao factorization (KRF), Procrustes estimation and Khatri-Rao factorization (ProKRaft), and the semi-algebraic framework for approximate canonical polyadic decomposition via simultaneous matrix diagonalization (SECSI), respectively. These approaches use filtering, parameter estimation and filtering, iterative algebraic factor matrix estimation and filtering, and algebraic factor matrix estimation, respectively. We propose two processing approaches to time-delay estimation (TDE). The first is batch processing taking data from several signal periods. Using batch processing we propose two algebraic approaches to TDE, in which diagonalization is achieved using the generalized eigenvalue decomposition (GEVD) of the first two frontal slices of the measurement tensor’s core tensor, estimated via MLSVD. The former approach, like the cited methods, in which simulations were performed with 1 LOS component and 1 NLOS component, and thus the measured data has full-rank tensor in all its modes, makes no assumption about the rank of the measurement tensor. The latter approach assumes scenarios in which more than 1 NLOS component is present and these are clustered, thus several vectors of one of the factor matrices which forms the tensor data are highly correlated, resulting in a rank-deficient measurement tensor in at least one mode. These proposed algebraic tensor-based schemes utilize the canonical polyadic decomposition via generalized eigenvalue decomposition (CPD-GEVD) and the decomposition in multilinear rank-(Lr, Lr, 1) terms via generalized eigenvalue decomposition ((Lr, Lr, 1)-GEVD) in order to improve the TDE of the LOS component in challenging scev narios. The second approach is adaptive processing of individual samples utilizing subspace tracking to iteratively estimate the subspace at each epoch. Using adaptive processing we propose two approaches, one applying FBA and ESPS to the data and estimating a higherorder eigenfilter, and the other using a parametric approach using DoA estimation. By extending the data model for an uniform rectangular array, we have a data stream of third-order tensors. For this model we propose three approaches to TDE based on HOE, CPD-GEVD, and standard tensor ESPRIT, respectively and employing a sequential truncation strategy to reduce the amount of operations necessary for each tensor mode