## Tensor Decompositions and Algorithms for Efficient Multidimensional Signal Processing (2024)

Compressed sensing approaches to large-scale tensor decompositions

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, ...

Vervliet, Nico — KU Leuven

Advanced Algebraic Concepts for Efficient Multi-Channel Signal Processing

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 ...

Roemer, Florian — Ilmenau University of Technology

Advanced Signal Processing Concepts for Multi-Dimensional Communication Systems

The widespread use of mobile internet and smart applications has led to an explosive growth in mobile data traffic. With the rise of smart homes, smart buildings, and smart cities, this demand is ever growing since future communication systems will require the integration of multiple networks serving diverse sectors, domains and applications, such as multimedia, virtual or augmented reality, machine-to-machine (M2M) communication / the Internet of things (IoT), automotive applications, and many more. Therefore, in the future, the communication systems will not only be required to provide Gbps wireless connectivity but also fulfill other requirements such as low latency and massive machine type connectivity while ensuring the quality of service. Without significant technological advances to increase the system capacity, the existing telecommunications infrastructure will be unable to support these multi-dimensional requirements. This poses an important demand for suitable waveforms with ...

Cheema, Sher Ali — Technische Universität Ilmenau

Explicit and implicit tensor decomposition-based algorithms and applications

Various real-life data such as time series and multi-sensor recordings can be represented by vectors and matrices, which are one-way and two-way arrays of numerical values, respectively. Valuable information can be extracted from these measured data matrices by means of matrix factorizations in a broad range of applications within signal processing, data mining, and machine learning. While matrix-based methods are powerful and well-known tools for various applications, they are limited to single-mode variations, making them ill-suited to tackle multi-way data without loss of information. Higher-order tensors are a natural extension of vectors (first order) and matrices (second order), enabling us to represent multi-way arrays of numerical values, which have become ubiquitous in signal processing and data mining applications. By leveraging the powerful utitilies offered by tensor decompositions such as compression and uniqueness properties, we can extract more information from multi-way ...

Boussé, Martijn — KU Leuven

Subspace-based exponential data fitting using linear and multilinear algebra

The exponentially damped sinusoidal (EDS) model arises in numerous signal processing applications. It is therefore of great interest to have methods able to estimate the parameters of such a model in the single-channel as well as in the multi-channel case. Because such a model naturally lends itself to subspace representation, powerful matrix approaches like HTLS in the single-channel case, HTLSstack in the multi-channel case and HTLSDstack in the decimative case have been developed to estimate the parameters of the underlying EDS model. They basically consist in stacking the signal in Hankel (single-channel) or block Hankel (multi- channel) data matrices. Then, the signal subspace is estimated by means of the singular value decomposition (SVD). The parameters of the model, namely the amplitudes, the phases, the damping factors, and the frequencies, are estimated from this subspace. Note that the sample covariance matrix ...

Papy, Jean-Michel — Katholieke Universiteit Leuven

Advanced Multi-Dimensional Signal Processing for Wireless Systems

The thriving development of wireless communications calls for innovative and advanced signal processing techniques targeting at an enhanced performance in terms of reliability, throughput, robustness, efficiency, flexibility, etc.. This thesis addresses such a compelling demand and presents new and intriguing progress towards fulfilling it. We mainly concentrate on two advanced multi-dimensional signal processing challenges for wireless systems that have attracted tremendous research attention in recent years, multi-carrier Multiple-Input Multiple-Output (MIMO) systems and multi-dimensional harmonic retrieval. As the key technologies of wireless communications, the numerous benefits of MIMO and multi-carrier modulation, e.g., boosting the data rate and improving the link reliability, have long been identified and have ignited great research interest. In particular, the Orthogonal Frequency Division Multiplexing (OFDM)-based multi-user MIMO downlink with Space-Division Multiple Access (SDMA) combines the twofold advantages of MIMO and multi-carrier modulation. It is the essential element ...

Cheng, Yao — Ilmenau University of Technology

Tensor-based blind source separation for structured EEG-fMRI data fusion

A complex physical system like the human brain can only be comprehended by the use of a combination of various medical imaging techniques, each of which shed light on only a specific aspect of the neural processes that take place beneath the skull. Electroencephalography (EEG) and functional magnetic resonance (fMRI) are two such modalities, which enable the study of brain (dys)function. While the EEG is measured with a limited set of scalp electrodes which record rapid electrical changes resulting from neural activity, fMRI offers a superior spatial resolution at the expense of only picking up slow fluctuations of oxygen concentration that takes place near active brain cells. Hence, combining these very complementary modalities is an appealing, but complicated task due to their heterogeneous nature. In this thesis, we devise advanced signal processing techniques which integrate the multimodal data stemming from ...

Van Eyndhoven, Simon — KU Leuven

Functional Neuroimaging Data Characterisation Via Tensor Representations

The growing interest in neuroimaging technologies generates a massive amount of biomedical data that exhibit high dimensionality. Tensor-based analysis of brain imaging data has by now been recognized as an effective approach exploiting its inherent multi-way nature. In particular, the advantages of tensorial over matrix-based methods have previously been demonstrated in the context of functional magnetic resonance imaging (fMRI) source localization; the identification of the regions of the brain which are activated at specific time instances. However, such methods can also become ineffective in realistic challenging scenarios, involving, e.g., strong noise and/or significant overlap among the activated regions. Moreover, they commonly rely on the assumption of an underlying multilinear model generating the data. In the first part of this thesis, we aimed at investigating the possible gains from exploiting the 3-dimensional nature of the brain images, through a higher-order tensorization ...

Christos Chatzichristos — National and Kapodistrian University of Athens

Learning from structured EEG and fMRI data supporting the diagnosis of epilepsy

Epilepsy is a neurological condition that manifests in epileptic seizures as a result of an abnormal, synchronous activity of a large group of neurons. Depending on the affected brain regions, seizures produce various severe clinical symptoms. Epilepsy cannot be cured and in many cases is not controlled by medication either. Surgical resection of the region responsible for generating the epileptic seizures might offer remedy for these patients. Electroencephalography (EEG) and functional magnetic resonance imaging (fMRI) measure the changes of brain activity in time over different locations of the brain. As such, they provide valuable information on the nature, the timing and the spatial origin of the epileptic activity. Unfortunately, both techniques record activity of different brain and artefact sources as well. Hence, EEG and fMRI signals are characterised by low signal to noise ratio. Data quality and the vast amount ...

Hunyadi, Borbála — KU Leuven

Decomposition methods with applications in neuroscience

The brain is the most important signal processing unit in the human body. It is responsible for receiving, processing and storing information. One of the possibilities to study brain functioning is by placing electrodes on the scalp and recording the synchronous neuronal activity of the brain. Such a recording measures a combination of active processes in the whole brain. Unfortunately, it is also contaminated by artifacts. By extracting the artifacts and removing them, cleaned recordings can be investigated. Furthermore, it is easier to look at speciﬁc brain activities, like an epileptic seizure, than at a combination. In this thesis, we present diﬀerent mathematical techniques that can be used to extract individual contributing sources from the measured signals for this purpose. We focused on Canonical Correlation Analysis (CCA), Independent Component Analysis (ICA) and Canonical/ Parallel Factor Analysis (CPA). We show that ...

De Vos, Maarten — Katholieke Universiteit Leuven

Nonnegative Matrix and Tensor Factorizations: Models, Algorithms and Applications

In many fields, such as linear algebra, computational geometry, combinatorial optimization, analytical chemistry and geoscience, nonnegativity of the solution is required, which is either due to the fact that the data is physically nonnegative, or that the mathematical modeling of the problem requires nonnegativity. Image and audio processing are two examples for which the data are physically nonnegative. Probability and graph theory are examples for which the mathematical modeling requires nonnegativity. This thesis is about the nonnegative factorization of matrices and tensors: namely nonnegative matrix factorization (NMF) and nonnegative tensor factorization (NTF). NMF problems arise in a wide range of scenarios such as the aforementioned fields, and NTF problems arise as a generalization of NMF. As the title suggests, the contributions of this thesis are centered on NMF and NTF over three aspects: modeling, algorithms and applications. On the modeling ...

Ang, Man Shun — Université de Mons

Advanced Algorithms for Polynomial Matrix Eigenvalue Decomposition

Matrix factorisations such as the eigen- (EVD) or singular value decomposition (SVD) offer optimality in often various senses to many narrowband signal processing algorithms. For broadband problems, where quantities such as MIMO transfer functions or cross spectral density matrices are conveniently described by polynomial matrices, such narrowband factorisations are suboptimal at best. To extend the utility of EVD and SVD to the broadband case, polynomial matrix factorisations have gained momen- tum over the past decade, and a number of iterative algorithms for particularly the polynomial matrix EVD (PEVD) have emerged. Existing iterative PEVD algorithms produce factorisations that are computationally costly (i) to calculate and (ii) to apply. For the former, iterative algorithms at every step eliminate off-diagonal energy, but this can be a slow process. For the latter, the polynomial order of the resulting factors, directly impacting on the implementa- ...

Corr, Jamie — University of Strathclyde

Improving data-driven EEG-FMRI analyses for the study of cognitive functioning

Understanding the cognitive processes that are going on in the human brain, requires the combination of several types of observations. For this reason, since several years, neuroscience research started to focus on multimodal approaches. One such multimodal approach is the combination of electroencephalography (EEG) and functional magnetic resonance imaging (fMRI). The non-invasive character of these two modalities makes their combination not only harmless and painless, but also especially suited for widespread research in both clinical and experimental applications. Moreover, the complementarity between the high temporal resolution of the EEG and the high spatial resolution of the fMRI, allows obtaining a more complete picture of the processes under study. However, the combination of EEG and fMRI is challenging, not only on the level of the data acquisition, but also when it comes to extracting the activity of interest and interpreting the ...

Vanderperren, Katrien — KU Leuven

Second-Order Multidimensional Independent Component Analysis: Theory and Methods

Independent component analysis (ICA) and blind source separation (BSS) deal with extracting a number of mutually independent elements from a set of observed linear mixtures. Motivated by various applications, this work considers a more general and more flexible model: the sources can be partitioned into groups exhibiting dependence within a given group but independence between two different groups. We argue that this is tantamount to considering multidimensional components, as opposed to the standard ICA case which is restricted to one-dimensional components. In this work, we focus on second-order methods to separate statistically-independent multidimensional components from their linear instantaneous mixture. The purpose of this work is to provide theoretical answers to questions which so far have been discussed mainly in the empirical domain. Namely, we provide a closed-form expression for the figure of merit, the mean square error (MSE), for multidimensional ...

Lahat, Dana — Tel Aviv University

Estima e Igualacion Ciega de Canales MIMO con y sin Redudancia Espacial (title in Spanish)

The majority of communication systems need the previous knowledge of the channel, which is usually estimated by means of a training sequence. However, the transmission of pilot symbols provokes a reduction in bandwidth efficiency, which precludes the system from reaching the limits predicted by the Information Theory. This problem has motivated the development of a large number of blind channel estimation and equalization techniques, which are able to obtain the channel or the source without the need of transmitting a training signal. Usually, these techniques are based on the previous knowledge of certain properties of the signal, such as its belonging to a finite alphabet, or its higher-order statistics. However, in the case of multiple-input multipleoutput (MIMO) systems, it has been proven that the second-order statistics of the observations provide the sufficient information for solving the blind problem. The aim ...

Rodriguez, Javier Via — Universidad de Cantabria

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