Materialien wie Kristalle, biologisches Gewebe oder elektroaktive Polymere kommen häufig in verschiedenen Anwendung, wie dem Prothesenbau oder der Simulation von künstlicher Muskulatur vor. Diese und viele weitere Materialien haben gemeinsam, dass sie unter gewissen Umständen ihre Form und andere Materialeigenschaften ändern. Um diese Veränderung beschreiben zu können, werden, abhängig von der Anwendung, verschiedene Tensoren unterschiedlicher Ordnung benutzt. Durch die Komplexität und die starke Abhängigkeit der Tensorbedeutung von der Anwendung, gibt es bisher kein Verfahren Tensoren höherer Ordnung darzustellen, welches standardmäßig benutzt wird. Auch bezogen auf einzelne Anwendungen gibt es nur sehr wenig Arbeiten, die sich mit der visuellen Darstellung dieser Tensoren auseinandersetzt. Diese Arbeit beschäftigt sich mit diesem Problem. Es werden drei verschiedene Methoden präsentiert, Tensoren höherer Ordnung zu analysieren und zu visualisieren. Alle drei Methoden basieren auf der sogenannte deviatorischen Zerlegung und der Multipoldarstellung. Mit Hilfe der Multipole können die Symmetrien des Tensors und damit des beschriebenen Materials bestimmt werden. Diese Eigenschaft wird in für die Visualisierung des Steifigkeitstensors benutzt. Die zweite Methode basiert direkt auf den Multipolen und kann damit beliebige Tensoren in drei Dimensionen darstellen. Dieses Verfahren wird anhand des Kopplungs Tensors, ein Tensor dritter Ordnung, vorgestellt. Die ersten zwei Verfahren sind lokale Glyph-basierte Verfahren. Das dritte Verfahren ist ein erstes globales Tensorvisualisierungsverfahren, welches Tensoren beliebiger Ordnung und Symmetry in drei Dimensionen mit Hilfe eines linienbasierten Verfahrens darstellt. / Materials like crystals, biological tissue or electroactive polymers are frequently used in applications like prosthesis construction or the simulation of artificial musculature. These and many other materials have in common that they change their shape and other material properties under certain circumstances. To describe these changes, different tensors of different order, dependent of the application, are used. Due to the complexity and the strong dependency of the tensor meaning of the application, there is, by now, no visualization method that is used by default. Also for specific applications there are only a few methods that address the visual analysis of higher-order tensors. This work adresses this problem. Three different methods to analyse and visualize tensors of higher order will be provided. All three methods are based on the so called deviatoric decomposition and the multipole representation. Using the multipoles the symmetries of a tensor and, therefore, of the described material, can be calculated. This property is used to visualize the stiffness tensor. The second method uses the multipoles directly and can be used for each tensor of any order in three dimensions. This method is presented by analysing the third-order coupling tensor. These two techniques are glyph-based visualization methods. The third one, a line-based method, is, according to our knowledge, a first global visualization method that can be used for an arbitrary tensor in three dimensions.

How to handle large multi-dimensional datasets such as hyperspectral images and video information both efficiently and effectively plays an important role in big-data processing. The characteristics of tensor low-rank decomposition in recent years demonstrate the importance of capturing the tensor structure adequately which usually yields efficacious approaches. In this dissertation, we first aim to explore the tensor singular value decomposition (t-SVD) with the nonconvex regularization on the multi-view subspace clustering (MSC) problem, then develop two new tensor decomposition models with the Bayesian inference framework on the tensor completion and tensor robust principal component analysis (TRPCA) and tensor completion (TC) problems. Specifically, the following developments for multi-dimensional datasets under the mathematical tensor framework will be addressed. (1) By utilizing the t-SVD proposed by Kilmer et al. \cite{kilmer2013third}, we unify the Hyper-Laplacian (HL) and exclusive $\ell_{2,1}$ (L21) regularization with Tensor Log-Determinant Rank Minimization (TLD) to identify data clusters from the multiple views' inherent information. Whereby the HL regularization maintains the local geometrical structure that makes the estimation prune to nonlinearities, and the mixed $\ell_{2,1}$ and $\ell_{1,2}$ regularization provides the joint sparsity within-cluster as well as the exclusive sparsity between-cluster. Furthermore, a log-determinant function is used as a tighter tensor rank approximation to discriminate the dimension of features. (2) By considering a tube as an atom of a third-order tensor and constructing a data-driven learning dictionary from the observed noisy data along the tubes of a tensor, we develop a Bayesian dictionary learning model with tensor tubal transformed factorization to identify the underlying low-tubal-rank structure of the tensor substantially with the data-adaptive dictionary for the TRPCA problem. With the defined page-wise operators, an efficient variational Bayesian dictionary learning algorithm is established for TPRCA that enables to update of the posterior distributions along the third dimension simultaneously. (3) With the defined matrix outer product into the tensor decomposition process, we present a new decomposition model for a third-order tensor. The fundamental idea is to decompose tensors mathematically in a compact manner as much as possible. By incorporating the framework of Bayesian probabilistic inference, the new tensor decomposition model on the subtle matrix outer product (BPMOP) is developed for the TC and TRPCA problems. Extensive experiments on synthetic data and real-world datasets are conducted for the multi-view clustering, TC, and TRPCA problems to demonstrate the desirable effectiveness of the proposed approaches, by detailed comparison with currently available results in the literature.

The ever-increasing applications of Big Data in improving networking application performancehave motivated the networking community to deploy it in SDN (Software defined network) toconstruct flexible, scalable, self-aware, and self-managing networks. The primary purpose ofthis research is to investigate the validity of tensor-decomposition, a well-knownmathematical approach for data reduction, to catch patterns in network traffic as an initialstep toward the network's intelligence.Using only three-dimensional (cubic) tensors (Source, Destination, Bandwidth). Theconducted research used both offline (not simulated) and online (Mininet and RYU controllersimulation) network traffic of the GEANT (TOTEM) dataset. From the tensor decompositionanalysis on the adjacency matricies, we caught traffic intensity patterns between nodes(switches), which provided suggestions that helps rebuild the topology (which nodes shouldbe physically connected to the others). However, capturing the patterns in the time revolutionwas invalid due to limitations in the three-dimensional tensor.

CoordenaÃÃo de AperfeiÃoamento de NÃvel Superior / In this dissertation, the face recognition problem is investigated from the standpoint of multilinear algebra, more specifically the tensor decomposition, and by making use of Gabor wavelets. The feature extraction occurs in two stages: first the Gabor wavelets are applied holistically in feature selection; Secondly facial images are modeled as a higher-order tensor according to the multimodal factors present. Then, the HOSVD is applied to separate the multimodal factors of the images. The proposed facial recognition approach exhibits higher average success rate and stability when there is variation in the various multimodal factors such as facial position, lighting condition and facial expression. We also propose a systematic way to perform cross-validation on tensor models to estimate the error rate in face recognition systems that explore the nature of the multimodal ensemble. Through the random partitioning of data organized as a tensor, the mode-n cross-validation provides folds as subtensors extracted of the desired mode, featuring a stratified method and susceptible to repetition of cross-validation with different partitioning. / Nesta dissertaÃÃo o problema de reconhecimento facial à investigado do ponto de vista da Ãlgebra multilinear, mais especificamente por meio de decomposiÃÃes tensoriais fazendo uso das wavelets de Gabor. A extraÃÃo de caracterÃsticas ocorre em dois estÃgios: primeiramente as wavelets de Gabor sÃo aplicadas de maneira holÃstica na seleÃÃo de caracterÃsticas; em segundo as imagens faciais sÃo modeladas como um tensor de ordem superior de acordo com o fatores multimodais presentes. Com isso aplicamos a decomposiÃÃo tensorial Higher Order Singular Value Decomposition (HOSVD) para separar os fatores que influenciam na formaÃÃo das imagens. O mÃtodo de reconhecimento facial proposto possui uma alta taxa de acerto e estabilidade quando hà variaÃÃo nos diversos fatores multimodais, tais como, posiÃÃo facial, condiÃÃo de iluminaÃÃo e expressÃo facial. Propomos ainda uma maneira sistemÃtica para realizaÃÃo da validaÃÃo cruzada em modelos tensoriais para estimaÃÃo da taxa de erro em sistemas de reconhecimento facial que exploram a natureza multilinear do conjunto de imagens. AtravÃs do particionamento aleatÃrio dos dados organizado como um tensor, a validaÃÃo cruzada modo-n proporciona a criaÃÃo de folds extraindo subtensores no modo desejado, caracterizando um mÃtodo estratificado e susceptÃvel a repetiÃÃes da validaÃÃo cruzada com diferentes particionamentos.

abstract: It is increasingly common to see machine learning techniques applied in conjunction with computational modeling for data-driven research in neuroscience. Such applications include using machine learning for model development, particularly for optimization of parameters based on electrophysiological constraints. Alternatively, machine learning can be used to validate and enhance techniques for experimental data analysis or to analyze model simulation data in large-scale modeling studies, which is the approach I apply here. I use simulations of biophysically-realistic cortical neuron models to supplement a common feature-based technique for analysis of electrophysiological signals. I leverage these simulated electrophysiological signals to perform feature selection that provides an improved method for neuron-type classification. Additionally, I validate an unsupervised approach that extends this improved feature selection to discover signatures associated with neuron morphologies - performing in vivo histology in effect. The result is a simulation-based discovery of the underlying synaptic conditions responsible for patterns of extracellular signatures that can be applied to understand both simulation and experimental data. I also use unsupervised learning techniques to identify common channel mechanisms underlying electrophysiological behaviors of cortical neuron models. This work relies on an open-source database containing a large number of computational models for cortical neurons. I perform a quantitative data-driven analysis of these previously published ion channel and neuron models that uses information shared across models as opposed to information limited to individual models. The result is simulation-based discovery of model sub-types at two spatial scales which map functional relationships between activation/inactivation properties of channel family model sub-types to electrophysiological properties of cortical neuron model sub-types. Further, the combination of unsupervised learning techniques and parameter visualizations serve to integrate characterizations of model electrophysiological behavior across scales. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020

The aging population requires new and innovative approaches to monitor and supervise medical and physical conditions in residential environments. For this purpose, various sensor and hardware systems are being developed by researchers and industrial companies. One way to monitor health status is the electrocardiogram (ECG), which noninvasively measures heart activity on the body surface. These measurements provide a simple and easy way to monitor health on a continuous basis. However, the use of ECG measurements outside a confined clinical setting, beyond purely medical purposes, is associated with considerable disadvantages resulting from the given freedom of movement. In this work, a substantial noise source in mobile ECG is examined: Motion artifacts. We study the spectral characteristics of motion artifacts for a set of different motions representing everyday activities, namely: standing up, bending forward, walking, running, jumping, and climbing stairs. Furthermore, we investigate to what extent the reference sensors (accelerometer, gyroscope, and skin-electrode impedance) are able to characterize and remove the recorded motion artifacts from the measurements. Our results demonstrate that motion artifacts markedly change their characteristics with a change in motion. While lowintensity movements manifest in lower frequency bands, higher intensity exercises provoke motion artifacts that are much more complex in their composition. These characteristics are correspondingly reflected in the correlation between reference sensors and artifacts. To overcome the drawbacks of motion artifacts in mobile measurements, we propose the application of tensor decomposition using canonical polyadic decomposition (CPD) as an example. A significant advantage of tensor factorization is that it can decompose the data without artificial constraints, unlike matrix factorization. We use CPD along with measurements obtained from different reference sensors to remove the artifacts. Wavelet transformation is utilized to transform ECG and reference data from vector to matrix format. Subsequently, a tensor is constructed by combining the heterogeneous measurements into a three-dimensional tensor. In this way, it is possible to access temporal and spectral features within the data simultaneously. Subsequently, we propose a methodology to predict the decomposition rank based on statistical features in the ECG that quantify the signal quality. To evaluate the performance of the decomposition process, we combine isolated motion artifacts recorded at the back with ECG obtained in rest to generate artificially corrupted data. The results suggest that CPD successfully removes motion artifacts from the data for all reference sensors regarded.

Les réseaux tensoriels offrent un cadre puissant pour représenter efficacement des objets de très haute dimension. Les réseaux tensoriels ont récemment montré leur potentiel pour les applications d’apprentissage automatique et offrent une vue unifiée des modèles de décomposition tensorielle courants tels que Tucker, tensor train (TT) et tensor ring (TR). Cependant, l’identification de la meilleure structure de réseau tensoriel à partir de données pour une tâche donnée est un défi. Dans cette thèse, nous nous appuyons sur le formalisme des réseaux tensoriels pour développer un algorithme adaptatif générique et efficace pour apprendre conjointement la structure et les paramètres d’un réseau de tenseurs à partir de données. Notre méthode est basée sur une approche simple de type gloutonne, partant d’un tenseur de rang un et identifiant successivement les bords du réseau tensoriel les plus prometteurs pour de petits incréments de rang. Notre algorithme peut identifier de manière adaptative des structures avec un petit nombre de paramètres qui optimisent efficacement toute fonction objective différentiable. Des expériences sur des tâches de décomposition de tenseurs, de complétion de tenseurs et de compression de modèles démontrent l’efficacité de l’algorithme proposé. En particulier, notre méthode surpasse l’état de l’art basée sur des algorithmes évolutionnaires introduit dans [26] pour la décomposition tensorielle d’images (tout en étant plusieurs ordres de grandeur plus rapide) et trouve des structures efficaces pour compresser les réseaux neuronaux en surpassant les approches populaires basées sur le format TT [30]. / Tensor Networks (TN) offer a powerful framework to efficiently represent very high-dimensional objects. TN have recently shown their potential for machine learning applications and offer a unifying view of common tensor decomposition models such as Tucker, tensor train (TT) and tensor ring (TR). However, identifying the best tensor network structure from data for a given task is challenging. In this thesis, we leverage the TN formalism to develop a generic and efficient adaptive algorithm to jointly learn the structure and the parameters of a TN from data. Our method is based on a simple greedy approach starting from a rank one tensor and successively identifying the most promising tensor network edges for small rank increments. Our algorithm can adaptively identify TN structures with small number of parameters that effectively optimize any differentiable objective function. Experiments on tensor decomposition, tensor completion and model compression tasks demonstrate the effectiveness of the proposed algorithm. In particular, our method outperforms the state-of-the- art evolutionary topology search introduced in [26] for tensor decomposition of images (while being orders of magnitude faster) and finds efficient structures to compress neural networks outperforming popular TT based approaches [30].

The aim of this thesis is to investigate the tensor decomposition of structured tensors related to SL(n)-irreducible representations. Structured tensors are multilinear objects satisfying specific symmetry relations and their decompositions are of great interest in the applications. In this thesis we look for the decompositions of tensors belonging to irreducible representations of SL(n) into sum of elementary objects associated to points of SL(n)-rational hoogeneous varieties. This family includes Veronese varieties (symmetric tensors), Grassmann varieties (skew-symmetric tensors), and flag varieties. A classic tool to study the decomposition of symmetric tensors is the apolarity theory, which dates back to Sylvester. An analogous skew-symmetric apolarity theory for skew-symmetric tensors have been developed only few years ago. In this thesis we describe a global apolarity theory called Schur apolarity theory, which is suitable for tensors belonging to any irreducible representation of SL(n). Examples, properties and applications of such apolarity are studied with details and original results both in algebra and geoemtry are provided.

Additive manufacturing (AM) is a process of creating objects from 3D model data by adding layers of material. AM technologies present several advantages compared to traditional manufacturing technologies, such as producing less material waste and being capable of producing parts with greater geometric complexity. However, deficiencies in the printing process due to high process uncertainty can affect the microstructural properties of a fabricated part leading to defects. In metal AM, previous studies have linked defects in parts with melt pool temperature fluctuations, with the size of the melt pool and the scan pattern being key factors associated with part defects. Thus being able to adjust certain process parameters during a part's fabrication, and knowing when to adjust these parameters, is critical to producing reliable parts. To know when to effectively adjust these parameters it is necessary to have models that can both identify when a defect has occurred and forecast the behavior of the process to identify if a defect will occur. This study focuses on the development of accurate forecasting models of the melt pool temperature distribution. Researchers at Mississippi State University have collected in-situ pyrometer data of a direct laser deposition process which captures the temperature distribution of the melt pool. The high-dimensionality and noise of the data pose unique challenges in developing accurate forecasting models. To overcome these challenges, a tensor decomposition modeling framework is developed that can actively learn and adapt to new data. The framework is evaluated on two datasets which demonstrates its ability to generate accurate forecasts and adjust to new data.

Recognition of physical and mental responses to stress is important for stress assessment and management as its negative effects in health can be prevented or reduced. Wearable technology, mainly using electroencephalogram (EEG), provides information such as tracking fitness activity, disease detection, and monitoring neurologicalstates of individuals. However, the recording of EEG signals from a wearable device is inconvenient, expensive, and uncomfortable during normal daily activities. This study introduces the application of tensor decomposition of non-EEG data for visualizing and classifying neurological statuses with application to human stress recognition. The multimodal dataset of non-EEG physiological signals publicly available from the PhysioNet database was used for testing the proposed method. To visualize the biosignals in a low dimensional feature space, the multi-way factorization technique known as the PARAFAC was applied for feature extraction. Results show visualizations that well separate the four groups of neurological statuses obtained from twenty healthy subjects. The extracted features were then used for pattern classification. Two statistical classifiers, which are the multinomial logit regression(MLR) and linear discriminant analysis (LDA), were implemented. The results show that the MLR and LDA can identify the four neurological statuses with accuracies of 95% and 98.8%, respectively. This study suggests the potential application of tensor decomposition for the analysis of physiological measurements collected from multiple sensors. Moreover, the proposed study contributes to the advancement of wearable technology for human stress monitoring. With tensor decomposition of complex multi-sensor or multi-channel data, simple classification techniques can be employed to achieve similar results obtained using sophisticated machine-learning techniques.