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Researchers often find it challenging to simplify complex data sets for downstream analysis.The combination of multiple variables can lead to complexity, and organizing such data sets into a higher dimensional structure can be more intuitive. For these data sets with an inherent multi-modal structure, a variety of dimensionality reduction techniques have allowed researchers to explore and infer biological interactions more effectively. Higher-order dimensionality reduction techniques all serve to accomplish the same purpose - to reduce the original data set and recover meaningful and interpretable patterns. The CANDECOMP/ PARAFAC (CP) model, a frequent choice among researchers for its interpretability, still requires metrics for validating its performance and assuring an appropriate model complexity is selected. While a common benchmark for these methods’ validation is typically the total residual error, imputation error (prediction error) can serve as a more trusted alternative. We describe an algorithm for fitting the PARAFAC model, censored alternating least squares, that innately handles missing values and compare it amongst alternating least squares and direct optimization using simulated and real data sets with varying degrees of missing values using these performance metrics. While each method has its own benefits, censored alternating least squares appears best suited for handling missing values, commonly present in the data that researchers look to investigate.

TensorView for MATLAB is a GUI-based visualization tool for depicting second-rank Cartesian tensors as surfaces on three-dimensional molecular models. Both ellipsoid and ovaloid tensor display formats are supported, and the software allows for easy conversion of Euler angles from common rotation schemes (active, passive, ZXZ, and ZYZ conventions) with visual feedback. In addition, the software displays all four orientation-equivalent Euler angle solutions for the placement of a single tensor in the molecular frame and can report relative orientations of two tensors with all 16 orientation-equivalent Euler angle sets that relate them. The salient relations are derived and illustrated through several examples. TensorView for MATLAB expands and complements the earlier implementation of TensorView within the Mathematica programming environment and can be run without a MATLAB license. TensorView for MATLAB is available through github at https://github.com/LeoSvenningsson/TensorViewforMatlab, and can also be accessed directly via the NMRbox resource.

Tensors or multidimensional arrays are higher order generalizations of ma-trices. They are natural structures for expressing data that have inherent higher order structures. Tensor decompositions and Tensor approximations play an important role in learning those hidden structures. They have many applications in machine learning, statistical learning, data science, signal processing, neuroscience, and more. Canonical Polyadic Decomposition (CPD) is a tensor decomposition that decomposes a tensor to minimal number of summation of rank 1 tensors. While for a given tensor, Low-Rank Tensor Approximation (LRTA) aims at finding a new one whose rank is small and that is close to the given one. We study the generating polynomials for computing tensor decompositions and low-rank approximations for given tensors and propose methods that can compute tensor decompositions for generic tensors under certain rank conditions. For low-rank tensor approximation, the proposed method guarantees that the constructed tensor is a good enough low-rank approximation if the tensor is to be approximated is close enough to a low-rank one. The proof built on perturbation analysis is presented.When the rank is higher than the second dimension, we are not able to findthe common zeros of generating polynomials directly. In this case, we need to use the quadratic equations that we get from those generating polynomials. We show that under certain conditions, we are able to find the tensor decompositions using standard linear algebra operations (i.e., solving linear systems, singular value decompositions, QR decompositions). Numerical examples and some comparisons are presented to show the performance of our algorithm. Multi-view learning is frequently used in data science. The pairwise correla- tion maximization is a classical approach for exploring the consensus of multiple views. Since the pairwise correlation is inherent for two views, the extensions to more views can be diversified and the intrin

In biology and bioinformatics, a variety of data share a common property that challenges numerous cutting-edge research studies: heterogeneities at the individual level with respect to more than one factor. Examples of such heterogeneities include but are not limited to: 1) unequal susceptibility of different patients, and 2) large diversity in gene length, GC content, etc., along with the resulting gene characteristics. For many biological data analysis studies, the critical first step is usually to infer null probability distribution of observed data with the heterogeneities in multiple (confounding) factors taken into account, so that we can further investigate the impact of other factor(s) of interest. Obviously, the heterogeneities heavily influence the potential conclusions that we may draw from statistical analyses of the data. However, modeling such heterogeneities has been challenging, not only due to the inapplicable explicit modeling of all factors with heterogeneous effects on the data, but also because of the non-independence of many factors from one another. Existing methods, either partially/fully neglected the heterogeneity issue at all, or took care of each factor's heterogeneity in isolation. Evidences have shown the insufficiency of such strategies and the errors they may produce in downstream analyses. The emergence of large-scale data sets provides the opportunity to directly and comprehensively learn the heterogeneity from the data without explicitly modeling the mechanisms behind or exerting strong assumptions. The data, as often stored or organized as multidimensional contingency tensors, lead to a natural perspective of modeling heterogeneity with each impact factor of interest being one dimension. The heterogeneity in each factor's impact on the variable of interest can be captured by the marginal property of the data tensor with respect to the corresponding dimension. For instance, in a single-cell sequencing dataset, which can be organize

© 2014 American Physical Society. Financial support was provided by the Spanish MICINN through Projects No. FIS2011-30145-C03-01 and No. FIS2011-30145-C03-02 (with FEDER contribution), and by the Junta de Andalucía through Project No. FQM219. R. C-R. acknowledges support from CSIC through the JAE-predoc program, cofunded by FSE., It is commonly accepted that general relativity is the only solution to the consistency problem that appears when trying to build a theory of interacting gravitons (massless spin-2 particles). Padmanabhan’s 2008 thought-provoking analysis raised some concerns that are having resonance in the community. In this paper we present the self-coupling problem in detail and explicitly solve the infinite-iterations scheme associated with it for the simplest theory of a graviton field, which corresponds to an irreducible spin-2 representation of the Poincaré group. We make explicit the nonuniqueness problem by finding an entire family of solutions to the self-coupling problem. Then we show that the only resulting theory which implements a deformation of the original gauge symmetry happens to have essentially the structure of unimodular gravity. This makes plausible the possibility of a natural solution to the first cosmological constant problem in theories of emergent gravity. Later on, we change for the sake of completeness the starting free-field theory to Fierz-Pauli theory, an equivalent theory but with a larger gauge symmetry. We indicate how to carry out the infinite summation procedure in a similar way. Overall, we conclude that as long as one requires the (deformed) preservation of internal gauge invariance, one naturally recovers the structure of unimodular gravity or general relativity but in a version that explicitly shows the underlying Minkowski spacetime, in the spirit of Rosen’s flat-background bimetric theory., Spanish MICINN, Junta de Andalucía, CSIC, FSE, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2011 IOP Publishing Ltd. International Workshop on the Dark Side of the Universe (6º. 2010. Guanajuato, México). This work is supported in part by CONACYT, DOE grant DOE/DE-FG02-94ER40823, FPA 2005-02327 project (DGICYT, Spain), CAM/UCM 910309 project, and MICINN Consolider-Ingenio MULTIDARK CSD2009-00064., The inert doublet model is an extension of the Standard Model of Elementary Particles that is defined by the only addition of a second Higgs doublet without couplings to quarks or leptons. This minimal framework has been studied for many reasons. In particular, it has been suggested that the new degrees of freedom contained in this doublet can account for the Dark Matter of the Universe., Consejo Nacional de Ciencia y Tecnologia (CONACYT), México, U. S. Department of Energy (DOE), Dirección General de Investigación Científica y Técnica (DGICYT), MINECO, España, Ministerio de Ciencia e Innovación (MICINN), España, Programa Consolider-Ingenio, MULTIDARK (MICINN), España, Ministerio de Economía y Competitividad (MINECO), España, Universidad Complutense de Madrid (UCM), Comunidad Autónoma de Madrid (CAM), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2012 IOP Publishing Ltd and Sissa Medialab srl. We would like to thank prof. G. F. R. Ellis for drawing our attention to reference [10]. This work has been supported by MICINN (Spain) projects numbers FIS2011-23000, FPA2011-27853-C02-01 and Consolider-Ingenio MULTIDARK CSD2009-00064. FDA would like to thank Theoretical Physics I department, Complutense University of Madrid for providing all the required academic resources. AdlCD acknowledges financial support from NRF and URC research fellowships (South Africa). JARC acknowledges the kind hospitality of ACGC/University of Cape Town while elaborating the manuscript., We study the accelerated expansion of the Universe through its consequences on a congruence of geodesics. We make use of the Raychaudhuri equation which describes the evolution of the expansion rate for a congruence of timelike or null geodesics. In particular, we focus on the space-time geometry contribution to this equation. By straightforward calculation from the metric of a Robertson-Walker cosmological model, it follows that in an accelerated expanding Universe the space-time contribution to the Raychaudhuri equation is positive for the fundamental congruence, favoring a non-focusing of the congruence of geodesics. However, the accelerated expansion of the present Universe does not imply a tendency of the fundamental congruence to diverge. It is shown that this is in fact the case for certain congruences of timelike geodesics without vorticity. Therefore, the focusing of geodesics remains feasible in an accelerated expanding Universe. Furthermore, a negative contribution to the Raychaudhuri equation from space-time geometry which is usually interpreted as the manifestation of the attractive character of gravity is restored in an accelerated expanding Robertson-Walker space-time at high speeds., Ministerio de Ciencia e Innovación (MICINN), España, Programa Consolider-Ingenio (Proyecto MULTIDARK), National Research Foundation (NRF), Sudáfrica, URC, Sudáfrica, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

Tensors of order d may be seen as arrays of entries indexed by d indices. They naturally appear as data, and arrays in applications such as chemistry, food science, forensics, environmental analysis and many other fields. Extracting and visualizing the underlying features from tensors is an important source separation problem. This chapter first describes an important class of data mining methods for tensors, namely low-rank tensor approximations (CPD, Tucker3) in the case of order d=3. In such a case, striking differences already exist compared to low-rank approximations of matrices, which are tensors of order d=2. Constrained decompositions and coupled decompositions, which are important variants of tensor decompositions, are also discussed in detail, along with practical learning algorithms. Finally, tensor decompositions are illustrated as a tool for source separation in food sciences. In particular fluorescence spectroscopy, electrophoresis in gel, or chromatography especially coupled with mass spectrometry, are techniques where tensor decompositions are known to be useful. Some of the many other source separation problems that may be tackled with tensor decompositions are briefly discussed in the concluding remarks.

Unlike the flat-view matrix, tensor provides an elegant representation to preserve the structure of data in the multidimensional nature. The wide use of tensor-based approaches highlights the limitation of matrix-based methods and has generated increasing needs in multidimensional data analytics. Driven by such demands, tensor decomposition with automatic rank determination has recently emerged as a promising tool for image processing, wireless communications and neural network compression. However, unlike the success in matrix-based schemes of exploring the correlation for performance improvement, existing tensor-based methods fail to consider the correlation structure in the tensor. In this paper, we propose to exploit the tensor intra-dimension correlation within the framework of sparse Bayesian learning. We provide the sparsity-inducing probabilistic models under both canonical polyadic decomposition and Tucker decomposition to capture the correlation structure effectively. We derive an efficient model inference method with a fast convergence rate based on the expectation-maximization algorithm. Furthermore, a complexity reduction method is proposed to speed up the computation by exploiting several properties of the Kronecker product. We further extend the proposed framework for the incomplete tensor corrupted by sparse outliers to achieve effective tensor completion and outlier mitigation. Simulation results demonstrate the superior performance of the proposed framework over those ignoring the intra-dimension correlation in terms of accuracy and sparsity.

Clustering and closure coefficients are among the most widely applied indicators in the description of the topological structure of a network. Many distinct definitions have been proposed over time, particularly in the case of weighted networks, where the choice of the weight attributed to the triangles is a crucial aspect. In the present work, in the framework of weighted directed multilayer networks, we extend the classical clustering and closure coefficients through the introduction of the clumping coefficient, which generalizes them to incomplete triangles of any type. We then organize the class of these coefficients in a systematic taxonomy in the more general context of weighted directed multilayer networks. Such cohesion coefficients have also been adapted to the different scales that characterize a multilayer network, in order to grasp their structure from different perspectives. We also show how the tensor formalism allows incorporating the new definitions, as well as all those existing in the literature, in a single unified writing, in such a way that a suitable choice of the involved adjacency tensors allows obtaining each of them. Finally, through some applications to simulated networks, we show the effectiveness of the proposed coefficients in capturing different peculiarities of the network structure on different scales.

Tensors of order d may be seen as arrays of entries indexed by d indices. They naturally appear as data, and arrays in applications such as chemistry, food science, forensics, environmental analysis and many other fields. Extracting and visualizing the underlying features from tensors is an important source separation problem. This chapter first describes an important class of data mining methods for tensors, namely low-rank tensor approximations (CPD, Tucker3) in the case of order d=3. In such a case, striking differences already exist compared to low-rank approximations of matrices, which are tensors of order d=2. Constrained decompositions and coupled decompositions, which are important variants of tensor decompositions, are also discussed in detail, along with practical learning algorithms. Finally, tensor decompositions are illustrated as a tool for source separation in food sciences. In particular fluorescence spectroscopy, electrophoresis in gel, or chromatography especially coupled with mass spectrometry, are techniques where tensor decompositions are known to be useful. Some of the many other source separation problems that may be tackled with tensor decompositions are briefly discussed in the concluding remarks.

Passenger clustering based on trajectory records is essential for transportation operators. However, existing methods cannot easily cluster the passengers due to the hierarchical structure of the passenger trip information, including multiple trips within each passenger and multi-dimensional information about each trip. Furthermore, existing approaches rely on an accurate specification of the clustering number to start. Finally, existing methods do not consider spatial semantic graphs such as geographical proximity and functional similarity between the locations. In this paper, we propose a novel tensor Dirichlet Process Multinomial Mixture model with graphs, which can preserve the hierarchical structure of the multi-dimensional trip information and cluster them in a unified one-step manner with the ability to determine the number of clusters automatically. The spatial graphs are utilized in community detection to link the semantic neighbors. We further propose a tensor version of Collapsed Gibbs Sampling method with a minimum cluster size requirement. A case study based on Hong Kong metro passenger data is conducted to demonstrate the automatic process of cluster amount evolution and better cluster quality measured by within-cluster compactness and cross-cluster separateness. The code is available at https://github.com/bonaldli/TensorDPMM-G.

Tensor, a multi-dimensional data structure, has been exploited recently in the machine learning community. Traditional machine learning approaches are vector- or matrix-based, and cannot handle tensorial data directly. In this paper, we propose a tensor train (TT)-based kernel technique for the first time, and apply it to the conventional support vector machine (SVM) for high-dimensional image classification with very small number of training samples. Specifically, we propose a kernelized support tensor train machine that accepts tensorial input and preserves the intrinsic kernel property. The main contributions are threefold. First, we propose a TT-based feature mapping procedure that maintains the TT structure in the feature space. Second, we demonstrate two ways to construct the TT-based kernel function while considering consistency with the TT inner product and preservation of information. Third, we show that it is possible to apply different kernel functions on different data modes. In principle, our method tensorizes the standard SVM on its input structure and kernel mapping scheme. This reduces the storage and computation complexity of kernel matrix construction from exponential to polynomial. The validity proof and computation complexity of the proposed TT-based kernel functions are provided elaborately. Extensive experiments are performed on high-dimensional fMRI and color images datasets, which demonstrates the superiority of the proposed scheme compared with the state-of-the-art techniques., Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Team Kim Batselier

Tensor methods have been used successfully in modeling multi-aspect data and finding useful latent factors in various applications. These applications range from finding meaningful communities in social networks, detecting fake news, brain data analysis (EEG) and countless other applications in various domains like Chemometrics, Psychometrics, and Signal Processing.Despite the success of tensor methods in a wide variety of problems, the application of tensor methods typically entails a number of assumptions which, even though they may pertain only to a limited set of applications or to certain algorithms, are so pervasive that they are considered conventional. However, those assumptions may not be generalizable and, in fact, may hurt performance of tensor methods broadly. The main motivation behind this thesis is to revisit some of those conventional assumptions, propose methods to tackle problems arising from relaxing those assumptions, and observing the subsequent advantages of doing so. Below is the list of works covered in this thesis. Firstly, we focus on a domain specific problem, in which we create a richer feature space for hyperspectral pixel classifications. The next three projects focus on time-evolving graphs, specifically how latent factors evolve over time in streaming tensor decomposition and adaptive granularity in multi-aspect temporal data.1. Feature Space Explosion: In this work, we used tensor factorization to generate a richer feature space for pixel classification in hyperspectral images. We propose a feature explosion technique which maps the input space to a higher dimensional space, which is contrary to the typical low-rank factorization to a low-dimensional space. We propose an algorithm called ORION, which exploits the multi-linear structure of the 3-D hyperspectral tensor using tensor decomposition and generates a space which is more expressive than the original input space. Effectiveness of our method was demonstrated against traditional

In multimodal sentiment analysis, a model learns to predict one’s opinion towards a certain entity after observing data from multiple sources. As humans merge information from all five senses to read someone,the model needs a proper fusion mechanism to utilize the data from multiple sources. Tensor fusion networks have been proposed and have shown to perform well for multimodal sentiment analysis. However, it suffers from the curse of dimensionality because they use the full tensor format. As a solution, low-rank fusion networks have been proposed, but they require us to pre-determine the rank, which is an ill-posed problem. Instead, we propose adaptive-rank fusion networks that can learn the rank in a data-driven way. We compare the three models in terms of accuracy, complexity, and runtime. Then, we test how adaptive-rank fusion networks react to being trained with a subset of the original training dataset.

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.

It is well known that quaternion tensors play a vital role in some areas, such as colour image and signal processing. So, in this thesis, we mainly extend the results of tensor ranks for real and complex numbers to quaternions which is non-commutative. A tensor is defined as a multidimensional array. However, there are several ways to decompose a tensor. For example, the decomposi- tion of a tensor into the sum of rank-one tensors is often called CANDECOM, PARAFAC, or CANDECOM/PARAFAC(CP). The minimum number of rank- one tensors needed to represent a tensor is known as the rank of a tensor. It is used here to find the ranks of some quaternion tensors. Determining the tensor ranks is difficult, even for small-size tensors. Also, the tensor ranks may vary on the basis fields. Up to now, most published results on the tensor ranks are over the real and complex number fields, which are commutative. In nature, not all problems are commutative. Hence, understanding tensors and tensor ranks in non-commutative cases is also essential. Nevertheless, a limited number of papers have been published on tensor ranks in non-commutative cases. In this thesis, we describe some properties of quaternion tensors, discuss the characterization of quaternion tensor ranks, compute the tensor ranks for several quaternion tensors using some simple evaluation methods, and find the upper bounds of maximal ranks for some quaternion tensors.

Tensor decompositions, such as CANDECOMP/PARAFAC (CP), are widely used in a variety of applications, such as chemometrics, signal processing, and machine learning. A broadly used method for computing such decompositions relies on the Alternating Least Squares (ALS) algorithm. When the number of components is small, regardless of its implementation, ALS exhibits low arithmetic intensity, which severely hinders its performance and makes GPU offloading ineffective. We observe that, in practice, experts often have to compute multiple decompositions of the same tensor, each with a small number of components (typically fewer than 20), to ultimately find the best ones to use for the application at hand. In this article, we illustrate how multiple decompositions of the same tensor can be fused together at the algorithmic level to increase the arithmetic intensity. Therefore, it becomes possible to make efficient use of GPUs for further speedups; at the same time, the technique is compatible with many enhancements typically used in ALS, such as line search, extrapolation, and non-negativity constraints. We introduce the Concurrent ALS algorithm and library, which offers an interface to MATLAB, and a mechanism to effectively deal with the issue that decompositions complete at different times. Experimental results on artificial and real datasets demonstrate a shorter time to completion due to increased arithmetic intensity.

Tensor decompositions, such as CANDECOMP/PARAFAC (CP), are widely used in a variety of applications, such as chemometrics, signal processing, and machine learning. A broadly used method for computing such decompositions relies on the Alternating Least Squares (ALS) algorithm. When the number of components is small, regardless of its implementation, ALS exhibits low arithmetic intensity, which severely hinders its performance and makes GPU offloading ineffective. We observe that, in practice, experts often have to compute multiple decompositions of the same tensor, each with a small number of components (typically fewer than 20), to ultimately find the best ones to use for the application at hand. In this article, we illustrate how multiple decompositions of the same tensor can be fused together at the algorithmic level to increase the arithmetic intensity. Therefore, it becomes possible to make efficient use of GPUs for further speedups; at the same time, the technique is compatible with many enhancements typically used in ALS, such as line search, extrapolation, and non-negativity constraints. We introduce the Concurrent ALS algorithm and library, which offers an interface to MATLAB, and a mechanism to effectively deal with the issue that decompositions complete at different times. Experimental results on artificial and real datasets demonstrate a shorter time to completion due to increased arithmetic intensity.

Tensor decompositions, such as CANDECOMP/PARAFAC (CP), are widely used in a variety of applications, such as chemometrics, signal processing, and machine learning. A broadly used method for computing such decompositions relies on the Alternating Least Squares (ALS) algorithm. When the number of components is small, regardless of its implementation, ALS exhibits low arithmetic intensity, which severely hinders its performance and makes GPU offloading ineffective. We observe that, in practice, experts often have to compute multiple decompositions of the same tensor, each with a small number of components (typically fewer than 20), to ultimately find the best ones to use for the application at hand. In this article, we illustrate how multiple decompositions of the same tensor can be fused together at the algorithmic level to increase the arithmetic intensity. Therefore, it becomes possible to make efficient use of GPUs for further speedups; at the same time, the technique is compatible with many enhancements typically used in ALS, such as line search, extrapolation, and non-negativity constraints. We introduce the Concurrent ALS algorithm and library, which offers an interface to MATLAB, and a mechanism to effectively deal with the issue that decompositions complete at different times. Experimental results on artificial and real datasets demonstrate a shorter time to completion due to increased arithmetic intensity.

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.

Individual passenger travel patterns have significant value in understanding passenger’s behavior, such as learning the hidden clusters of locations, time, and passengers. The learned clusters further enable commercially beneficial actions such as customized services, promotions, data-driven urban-use planning, peak hour discovery, and so on. However, the individualized passenger modeling is very challenging for the following reasons: 1) The individual passenger travel data are multi-dimensional spatiotemporal big data, including at least the origin, destination, and time dimensions; 2) Moreover, individualized passenger travel patterns usually depend on the external environment, such as the distances and functions of locations, which are ignored in most current works. This work proposes a multi-clustering model to learn the latent clusters along the multiple dimensions of Origin, Destination, Time, and eventually, Passenger (ODT-P). We develop a graph-regularized tensor Latent Dirichlet Allocation (LDA) model by first extending the traditional LDA model into a tensor version and then applies to individual travel data. Then, the external information of stations is formulated as semantic graphs and incorporated as the Laplacian regularizations; Furthermore, to improve the model scalability when dealing with massive data, an online stochastic learning method based on tensorized variational Expectation-Maximization algorithm is developed. Finally, a case study based on passengers in the Hong Kong metro system is conducted and demonstrates that a better clustering performance is achieved compared to state-of-the-arts with the improvement in point-wise mutual information index and algorithm convergence speed by a factor of two. © 2022, The Author(s).

Tensor, a multi-dimensional data structure, has been exploited recently in the machine learning community. Traditional machine learning approaches are vector- or matrix-based, and cannot handle tensorial data directly. In this paper, we propose a tensor train (TT)-based kernel technique for the first time, and apply it to the conventional support vector machine (SVM) for high-dimensional image classification with very small number of training samples. Specifically, we propose a kernelized support tensor train machine that accepts tensorial input and preserves the intrinsic kernel property. The main contributions are threefold. First, we propose a TT-based feature mapping procedure that maintains the TT structure in the feature space. Second, we demonstrate two ways to construct the TT-based kernel function while considering consistency with the TT inner product and preservation of information. Third, we show that it is possible to apply different kernel functions on different data modes. In principle, our method tensorizes the standard SVM on its input structure and kernel mapping scheme. This reduces the storage and computation complexity of kernel matrix construction from exponential to polynomial. The validity proof and computation complexity of the proposed TT-based kernel functions are provided elaborately. Extensive experiments are performed on high-dimensional fMRI and color images datasets, which demonstrates the superiority of the proposed scheme compared with the state-of-the-art techniques., Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Team Kim Batselier

A wide range of practical problems involve computing multi-dimensional integrations. However, in most cases, it is hard to find analytical solutions to these multi-dimensional integrations. Their numerical solutions always suffer from the `curse of dimension', which means the computational complexity grows exponentially with respect to the dimension. There is an existing approach that approximates multivariate functions by a tensor of truncated multi-dimensional Fourier series coefficients, and uses the Stochastic Gradient Descent method to solve the lower-rank CPD model, which is used to reduce the computational complexity of the coefficient tensor. In contrast to this work, this thesis project extends its application to solve multi-dimensional integrations, utilizing the Fourier-cosine series expansion to represent the integrand. This project also replaces the SGD method with the Conjugate Gradient method, which improves the function matching accuracy significantly and also has great integration accuracy. The computational cost is reduced regardingly as well. This thesis also tests an expectation operator related to the COS method, which can be used to compute the expectation of functions of several random variables with much less computational complexity. This method filters out insignificant Fourier-cosine basis functions of the marginal distribution functions, and uses the selected `principal' basis functions to compute Fourier-cosine coefficients of the joint density function by the high-dimensional COS method. The test results show that more than half of the Fourier-cosine series terms can be dropped per dimension while the expectation accuracy is kept, and the correlation does not influence the expectation accuracy significantly for target functions of normally distributed random variables., Applied Mathematics

This report documents the program and the outcomes of Dagstuhl Seminar 22101 "Tensor Computations: Applications and Optimization". Tensors are higher dimensional analogs of matrices, and represent a key data abstraction for many applications in computational science and data science. Widely used shared infrastructure exists for linear algebra, while, in contrast, for tensor computations, there is no consensus on standard building blocks. This Dagstuhl Seminar aimed to bring together users, and performance optimization specialists, to build such foundations. We present the abstracts of the 5 tutorials and 14 talks given. The working groups and their outcomes so far are then presented.

En este proyecto se ha desarrollado un prototipo de prótesis biónica, creado en su totalidad desde 0, tanto en formato físico como la parte mecánica y electrónica. Han sido empleados métodos de análisis con softwares de simulación, diseño y métodos prácticos. El trabajo consta de una parte mecánica y una electrónica para su control. La parte mecánica incluye diferentes ámbitos adquiridos tanto de manera autodidacta como por la UPC. Se justifican los resultados y procesos seleccionados en los siguientes apartados del trabajo: Estructura mecánica de una prótesis biónica, Desarrollo del modelado, Ensamble, Estudio de fuerzas y Estudio de resistencia del material. Cada apartado contiene la información necesaria para entender el trabajo realizado. Se decide utilizar un software de simulación y diseño, para los estudios de fuerzas y estudio de resistencia del material, en su faceta como software de simulación y en el apartado Desarrollo del modelado, su faceta como software de diseño 3D. Para el desarrollo de la parte electrónica, se han escogido unos componentes acorde a las necesidades básicas del prototipo. Seguidamente se ha realizado un esquema, el cual se ha plasmado en una placa diseñada gracias a un programa especializado en ello e impresa por una empresa externa. El control de la prótesis se ha llevado a cabo gracias a un código programado y un microcontrolador que lo ejecuta. Se ha incluido un apartado denominado Montaje ya que se ha debido realizar una parte más práctica del trabajo, donde se procede a completar su ensamblado de todos los componentes y poder observar de manera real el prototipo creado. Se han realizado pruebas de funcionalidad y rendimiento e incorporado un apartado denominado como Mejoras para un posible prototipo final, con visión a una futura prótesis final de carácter comercial., En aquest projecte s'ha desenvolupat un prototip de pròtesi biònica, creat íntegrament des de 0, tant en format físic com la part mecànica i electrònica. Han estat emprats mètodes d'anàlisis amb softwares de simulació, disseny i mètodes pràctics. El treball consta d'una part mecànica i una electrònica per al seu control. La part mecànica inclou diferents àmbits adquirits tant de manera autodidacta com per la UPC. Es justifiquen els resultats i processos seleccionats en els següents apartats del treball: Estructura mecànica d'una pròtesi biònica, Desenvolupament del modelatge, Assembli, Estudi de forces i Estudi de resistència del material. Cada apartat conté la informació necessària per a entendre el treball realitzat. Es decideix utilitzar un programari de simulació i disseny, per als estudis de forces i estudi de resistència del material, en la seva faceta com a programari de simulació i en l'apartat Desenvolupament del modelatge, la seva faceta com a programari de disseny 3D. Per al desenvolupament de la part electrònica, s'han triat els components concorde a les necessitats bàsiques del prototip. Seguidament s'ha realitzat un esquema, el qual s'ha plasmat en una placa dissenyada gràcies a un programa especialitzat en això i impresa per una empresa externa. El control de la pròtesi s'ha dut a terme gràcies a un codi programat i un microcontrolador que l'executa. S'ha inclòs un apartat denominat Muntatge ja que s'ha hagut de realitzar una part més pràctica del treball, on es procedeix a completar el seu assemblat de tots els components i poder observar de manera real el prototip creat. S'han realitzat proves de funcionalitat i rendiment i incorporat un apartat denominat com a millores per a un possible prototip final, amb visió a una futura pròtesi final de caràcter comercial., In this project, a prototype bionic prosthesis has been developed, created entirely from scratch, both in physical format and the mechanical and electronic parts. Analysis methods have been used with simulation software, design and practical methods. The work consists of a mechanical part and an electronic part for its control. The mechanical part includes different areas acquired both self-taught and by the UPC. The selected results and processes are justified in the following sections of the work: Mechanical structure of a bionic prosthesis, Modelling development, Assembly, Study of forces and Study of material resistance. Each section contains the information necessary to understand the work done. It was decided to use simulation and design software for the force studies and material resistance study, in its facet as simulation software, and in the modelling development section, in its facet as 3D design software. For the development of the electronic part, components have been chosen according to the basic needs of the prototype. A schematic was then drawn up, which was then translated into a board designed thanks to a specialised programme and printed by an external company. The control of the prosthesis has been carried out thanks to a programmed code and a microcontroller that executes it. A section called Assembly has been included, as a more practical part of the work has been carried out, where the assembly of all the components is completed and the prototype created can be observed in a real way. Functionality and performance tests have been carried out and a section called Improvements for a possible final prototype has been included, with a view to a future final prosthesis of a commercial nature.

Big data and the use of advanced technologies are relevant topics in the financial market. In this context, complex networks became extremely useful in describing the structure of complex financial systems. In particular, the time evolution property of the stock markets have been described by temporal networks. However, these approaches fail to consider the interactions over time between assets. To overcome this drawback, financial markets can be described by multiplex networks where the different relations between nodes can be conveniently expressed structuring the network through different layers. To catch this kind of interconnections we provide new local clustering coefficients for multiplex networks, looking at the network from different perspectives depending on the node position, as well as a global clustering coefficient for the whole network. We also prove that all the well-known expressions for clustering coefficients existing in the literature, suitably extended to the multiplex framework, may be unified into our proposal. By means of an application to the multiplex temporal financial network, based on the returns of the S&P100 assets, we show that the proposed measures prove to be effective in describing dependencies between assets over time.

It is well known that quaternion tensors play a vital role in some areas, such as colour image and signal processing. So, in this thesis, we mainly extend the results of tensor ranks for real and complex numbers to quaternions which is non-commutative. A tensor is defined as a multidimensional array. However, there are several ways to decompose a tensor. For example, the decomposi- tion of a tensor into the sum of rank-one tensors is often called CANDECOM, PARAFAC, or CANDECOM/PARAFAC(CP). The minimum number of rank- one tensors needed to represent a tensor is known as the rank of a tensor. It is used here to find the ranks of some quaternion tensors. Determining the tensor ranks is difficult, even for small-size tensors. Also, the tensor ranks may vary on the basis fields. Up to now, most published results on the tensor ranks are over the real and complex number fields, which are commutative. In nature, not all problems are commutative. Hence, understanding tensors and tensor ranks in non-commutative cases is also essential. Nevertheless, a limited number of papers have been published on tensor ranks in non-commutative cases. In this thesis, we describe some properties of quaternion tensors, discuss the characterization of quaternion tensor ranks, compute the tensor ranks for several quaternion tensors using some simple evaluation methods, and find the upper bounds of maximal ranks for some quaternion tensors.

In this paper, we investigate effective sketching schemes via sparsification for high dimensional multilinear arrays or tensors. More specifically, we propose a novel tensor sparsification algorithm that retains a subset of the entries of a tensor in a judicious way, and prove that it can attain a given level of approximation accuracy in terms of tensor spectral norm with a much smaller sample complexity when compared with existing approaches. In particular, we show that for a kth order d × ... × d cubic tensor of stable rank rs, the sample size requirement for achieving a relative error ε is, up to a logarithmic factor, of the order r1/2s dk/2/ε when ε is relatively large, and rsd/ε2 and essentially optimal when ε is sufficiently small. It is especially noteworthy that the sample size requirement for achieving a high accuracy is of an order independent of k. To further demonstrate the utility of our techniques, we also study how higher order singular value decomposition (HOSVD) of large tensors can be efficiently approximated via sparsification. IEEE

Polynomial optimization considers optimization problems defined by polynomials. In contrast to classical nonlinearoptimization, it aims at finding global optimizers. Tensors are natural higher-order generalizations of matrices and are closely related to polynomials and moments. They are powerful tools in studying tensors. Many tensor problems can be formulated as polynomial optimization problems. We propose a complete semidefinite relaxation algorithmfor detecting the copositivity of a symmetric tensor. We show that the detection can be done by solving a finite number of semidefinite relaxations for all tensors.For the saddle point problem of polynomials,we give an algorithm for computing saddle points. We show that: i) if there exists a saddle point, our algorithm can get one by solving a finite number of Lasserre type semidefinite relaxations; ii) if there is no saddle point, our algorithm can detect its nonexistence. Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Canonical basis Hermitian tensors, real Hermitian tensors, special matrix flattenings, positive semidefiniteness, and separability are studied. We further study how to detect separability of Hermitian tensors. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it.The problem of learning diagonal Gaussian mixture models can be formulated as computing incomplete symmetric tensor decompositions. We use generating polynomials to compute incomplete symmetric tensor decompositions and approximations. Then the tensor approximation is used to learn diagonal Gaussian mixture models. When the first and third order moments are sufficiently accurate, we show that the obtained parameters for the Gaussian mixture models are also highly accurate.

Polynomial optimization considers optimization problems defined by polynomials. In contrast to classical nonlinearoptimization, it aims at finding global optimizers. Tensors are natural higher-order generalizations of matrices and are closely related to polynomials and moments. They are powerful tools in studying tensors. Many tensor problems can be formulated as polynomial optimization problems. We propose a complete semidefinite relaxation algorithmfor detecting the copositivity of a symmetric tensor. We show that the detection can be done by solving a finite number of semidefinite relaxations for all tensors.For the saddle point problem of polynomials,we give an algorithm for computing saddle points. We show that: i) if there exists a saddle point, our algorithm can get one by solving a finite number of Lasserre type semidefinite relaxations; ii) if there is no saddle point, our algorithm can detect its nonexistence. Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Canonical basis Hermitian tensors, real Hermitian tensors, special matrix flattenings, positive semidefiniteness, and separability are studied. We further study how to detect separability of Hermitian tensors. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it.The problem of learning diagonal Gaussian mixture models can be formulated as computing incomplete symmetric tensor decompositions. We use generating polynomials to compute incomplete symmetric tensor decompositions and approximations. Then the tensor approximation is used to learn diagonal Gaussian mixture models. When the first and third order moments are sufficiently accurate, we show that the obtained parameters for the Gaussian mixture models are also highly accurate.

Individualized passenger travel pattern is of significant research value since the abundant information from individual trajectory data could help discover the useful insights about the multi-clustering of origin, destination, time, etc., and the passenger cluster. However, this task is rather challenging, given the data is high-dimensional, with complex spatiotemporal structure, and exhibits sparse patterns. Moreover, individual travel patterns are also affected by external information, such as the distances and functions of locations and points of interest, which are ignored in most current works. To tackle these challenges, we proposed two novel frameworks based on topic models with the external information incorporated as graphs: Trips and passengers are formulated as tensor words and tensor documents to preserve data nature; To learn multiclustering, we proposed a graph-regularized tensor Latent Dirichlet Allocation model, with graph structure formulated as Laplacian penalty; To learn passenger clustering, we proposed a graph-based tensor Dirich-let Multinomial Mixture model with graph Laplacian penalty and l 1 -norm penalty for cluster amount auto-determination.

Identifying misinformation is one of the most challenging problems in today's interconnected world. The vast majority of the state-of-the-art in detecting misinformation are fully supervised, requiring a large number of high-quality human annotations. However, the availability of such annotations cannot be taken for granted, since it is very costly, time-consuming, and challenging to do so in a way that keeps up with the proliferation of misinformation. In this thesis, we are interested in exploring scenarios where the number of annotations is limited. In such scenarios, we leverage a multilinear framework a.k.a. "tensor" for a variety of modalities which has been shown to be interpretable and a proper tool for semi-supervised and unsupervised settings where there is a few or no annotation. In this dissertation, We propose a number of tensor-based techniques, organized in the following parts:Content-based techniques of misinformation detection. We propose a novel strategy mixing tensor-based content modeling and semi-supervised learning on article embeddings which requires very few labels. Driven by the effectiveness of our tensor embeddings, we propose a novel text augmentation framework i.e., Vec2Node leveraging tensor decomposition to generate synthetic samples by exploiting local and global information in text and reducing concept drift. Vec2Node leverages self-training from in-domain unlabeled data augmented with tensorized word embeddings. Finally. we propose a hybrid summarization framework that incorporates both extractive and abstractive techniques for capturing misinformative key phrases. Ensemble techniques for multi-aspect detection of misinformation We investigate how to tap into a diverse number of aspects that characterize a news article, can compensate for the lack of labels. We propose two tensor-based techniques for ensemble learning: HiJoD, a 2-level decomposition framework that leverages article content, context of social sharing behaviors, and h

Individualized passenger travel pattern is of significant research value since the abundant information from individual trajectory data could help discover the useful insights about the multi-clustering of origin, destination, time, etc., and the passenger cluster. However, this task is rather challenging, given the data is high-dimensional, with complex spatiotemporal structure, and exhibits sparse patterns. Moreover, individual travel patterns are also affected by external information, such as the distances and functions of locations and points of interest, which are ignored in most current works. To tackle these challenges, we proposed two novel frameworks based on topic models with the external information incorporated as graphs: Trips and passengers are formulated as tensor words and tensor documents to preserve data nature; To learn multiclustering, we proposed a graph-regularized tensor Latent Dirichlet Allocation model, with graph structure formulated as Laplacian penalty; To learn passenger clustering, we proposed a graph-based tensor Dirich-let Multinomial Mixture model with graph Laplacian penalty and l 1 -norm penalty for cluster amount auto-determination.

In this thesis two topics are discussed. The covariant formulation ofMaxwell’s equations of electromagnetism and the formulation of said equations in the context of a rotating frame of reference. Through the development of the necessary theories of Differential Geometry and Special Relativity we show how to formulateMaxwell’s equations in terms of the covariant derivative and in terms of the hodge operator and differential operator. Using this formulation we study the transformation properties of these equations under Lorentz transformations and we conclude that they remain invariant under said transformations. Secondly,we discuss literature concerning the problem of electromagnetism in the context of a rotating frame of reference. We show that the method of direct transformation to a rotating frame results in an unobservable frame of reference and we show that it is impossible to reconstruct Coriolis like forces in an observable frame of reference.

With the increasing number of wireless devices and the phenomenal amount of data that is being generated by them, there is a growing interest in the wireless communications community to complement the traditional model-driven design approaches with data-driven machine learning (ML)-based solutions. However, managing the large-scale multi-dimensional data to maintain the efficiency and scalability of the ML algorithms has obviously been a challenge. Tensors provide a useful framework to represent multi-dimensional data in an integrated manner by preserving relationships in data across different dimensions. This thesis studies two new applications of tensors to ML for wireless communications where the tensor structure of the concerned data is exploited in novel ways. The first contribution of this thesis is a tensor learning-based low-complexity precoder codebook design technique for a full-dimension multiple-input multiple-output (FD-MIMO) system with a uniform planar antenna (UPA) array at the transmitter (Tx) whose channel distribution is available through a dataset. Represented as a tensor, the FD-MIMO channel is further decomposed using a tensor decomposition technique to obtain an optimal precoder which is a function of Kronecker-Product (KP) of two low-dimensional precoders, each corresponding to the horizontal and vertical dimensions of the FD-MIMO channel. From the design perspective, we have made contributions in deriving a criterion for optimal product precoder codebooks using the obtained low-dimensional precoders. We show that this product codebook design problem is an unsupervised clustering problem on a Cartesian Product Grassmann Manifold (CPM), where the optimal cluster centroids form the desired codebook. We further simplify this clustering problem to a $K$-means algorithm on the low-dimensional factor Grassmann manifolds (GMs) of the CPM which correspond to the horizontal and vertical dimensions of the UPA, thus significantly reducing the complexity

Polynomial optimization considers optimization problems defined by polynomials. In contrast to classical nonlinearoptimization, it aims at finding global optimizers. Tensors are natural higher-order generalizations of matrices and are closely related to polynomials and moments. They are powerful tools in studying tensors. Many tensor problems can be formulated as polynomial optimization problems. We propose a complete semidefinite relaxation algorithmfor detecting the copositivity of a symmetric tensor. We show that the detection can be done by solving a finite number of semidefinite relaxations for all tensors.For the saddle point problem of polynomials,we give an algorithm for computing saddle points. We show that: i) if there exists a saddle point, our algorithm can get one by solving a finite number of Lasserre type semidefinite relaxations; ii) if there is no saddle point, our algorithm can detect its nonexistence. Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Canonical basis Hermitian tensors, real Hermitian tensors, special matrix flattenings, positive semidefiniteness, and separability are studied. We further study how to detect separability of Hermitian tensors. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it.The problem of learning diagonal Gaussian mixture models can be formulated as computing incomplete symmetric tensor decompositions. We use generating polynomials to compute incomplete symmetric tensor decompositions and approximations. Then the tensor approximation is used to learn diagonal Gaussian mixture models. When the first and third order moments are sufficiently accurate, we show that the obtained parameters for the Gaussian mixture models are also highly accurate.

Identifying misinformation is one of the most challenging problems in today's interconnected world. The vast majority of the state-of-the-art in detecting misinformation are fully supervised, requiring a large number of high-quality human annotations. However, the availability of such annotations cannot be taken for granted, since it is very costly, time-consuming, and challenging to do so in a way that keeps up with the proliferation of misinformation. In this thesis, we are interested in exploring scenarios where the number of annotations is limited. In such scenarios, we leverage a multilinear framework a.k.a. "tensor" for a variety of modalities which has been shown to be interpretable and a proper tool for semi-supervised and unsupervised settings where there is a few or no annotation. In this dissertation, We propose a number of tensor-based techniques, organized in the following parts:Content-based techniques of misinformation detection. We propose a novel strategy mixing tensor-based content modeling and semi-supervised learning on article embeddings which requires very few labels. Driven by the effectiveness of our tensor embeddings, we propose a novel text augmentation framework i.e., Vec2Node leveraging tensor decomposition to generate synthetic samples by exploiting local and global information in text and reducing concept drift. Vec2Node leverages self-training from in-domain unlabeled data augmented with tensorized word embeddings. Finally. we propose a hybrid summarization framework that incorporates both extractive and abstractive techniques for capturing misinformative key phrases. Ensemble techniques for multi-aspect detection of misinformation We investigate how to tap into a diverse number of aspects that characterize a news article, can compensate for the lack of labels. We propose two tensor-based techniques for ensemble learning: HiJoD, a 2-level decomposition framework that leverages article content, context of social sharing behaviors, and h

New relations involving the Riemann, Ricci and Einstein tensors that have to hold for a given geometry to admit Killing-Yano tensors (KYTs) are described. These relations are then used to introduce novel conserved 'currents' involving such KYTs. For a particular current based on the Einstein tensor, we discuss the issue of conserved charges and consider implications for matter coupled to gravity. The condition on the background geometry to allow asymptotic conserved charges for a current introduced by Kastor and Traschen is found and a number of other new aspects of this current are commented on. In particular we show that it vanishes for rank (D - 1) KYTs in D dimensions.

With the increasing number of wireless devices and the phenomenal amount of data that is being generated by them, there is a growing interest in the wireless communications community to complement the traditional model-driven design approaches with data-driven machine learning (ML)-based solutions. However, managing the large-scale multi-dimensional data to maintain the efficiency and scalability of the ML algorithms has obviously been a challenge. Tensors provide a useful framework to represent multi-dimensional data in an integrated manner by preserving relationships in data across different dimensions. This thesis studies two new applications of tensors to ML for wireless communications where the tensor structure of the concerned data is exploited in novel ways. The first contribution of this thesis is a tensor learning-based low-complexity precoder codebook design technique for a full-dimension multiple-input multiple-output (FD-MIMO) system with a uniform planar antenna (UPA) array at the transmitter (Tx) whose channel distribution is available through a dataset. Represented as a tensor, the FD-MIMO channel is further decomposed using a tensor decomposition technique to obtain an optimal precoder which is a function of Kronecker-Product (KP) of two low-dimensional precoders, each corresponding to the horizontal and vertical dimensions of the FD-MIMO channel. From the design perspective, we have made contributions in deriving a criterion for optimal product precoder codebooks using the obtained low-dimensional precoders. We show that this product codebook design problem is an unsupervised clustering problem on a Cartesian Product Grassmann Manifold (CPM), where the optimal cluster centroids form the desired codebook. We further simplify this clustering problem to a $K$-means algorithm on the low-dimensional factor Grassmann manifolds (GMs) of the CPM which correspond to the horizontal and vertical dimensions of the UPA, thus significantly reducing the complexity

Individualized passenger travel pattern is of significant research value since the abundant information from individual trajectory data could help discover the useful insights about the multi-clustering of origin, destination, time, etc., and the passenger cluster. However, this task is rather challenging, given the data is high-dimensional, with complex spatiotemporal structure, and exhibits sparse patterns. Moreover, individual travel patterns are also affected by external information, such as the distances and functions of locations and points of interest, which are ignored in most current works. To tackle these challenges, we proposed two novel frameworks based on topic models with the external information incorporated as graphs: Trips and passengers are formulated as tensor words and tensor documents to preserve data nature; To learn multiclustering, we proposed a graph-regularized tensor Latent Dirichlet Allocation model, with graph structure formulated as Laplacian penalty; To learn passenger clustering, we proposed a graph-based tensor Dirich-let Multinomial Mixture model with graph Laplacian penalty and l 1 -norm penalty for cluster amount auto-determination.

In this thesis two topics are discussed. The covariant formulation ofMaxwell’s equations of electromagnetism and the formulation of said equations in the context of a rotating frame of reference. Through the development of the necessary theories of Differential Geometry and Special Relativity we show how to formulateMaxwell’s equations in terms of the covariant derivative and in terms of the hodge operator and differential operator. Using this formulation we study the transformation properties of these equations under Lorentz transformations and we conclude that they remain invariant under said transformations. Secondly,we discuss literature concerning the problem of electromagnetism in the context of a rotating frame of reference. We show that the method of direct transformation to a rotating frame results in an unobservable frame of reference and we show that it is impossible to reconstruct Coriolis like forces in an observable frame of reference.

We develop a theory for the commutative combination of quantitative effects, their tensor, given as a combination of quantitative equational theories that imposes mutual commutation of the operations from each theory. As such, it extends the sum of two theories, which is just their unrestrained combination. Tensors of theories arise in several contexts; in particular, in the semantics of programming languages, the monad transformer for global state is given by a tensor. We show that under certain assumptions on the quantitative theories the free monad that arises from the tensor of two theories is the categorical tensor of the free monads on the theories. As an application, we provide the first algebraic axiomatizations of labelled Markov processes and Markov decision processes. Apart from the intrinsic interest in the axiomatizations, it is pleasing they are obtained compositionally by means of the sum and tensor of simpler quantitative equational theories.

PARAFAC2 is a useful algorithm for decomposing tensors that do not have low-rank variation such as e.g. PARAFAC requires. It has been applied in analyzing different types of multi-way data, such as GC-MS data. Since the optimization in fitting the loss function of PARAFAC2 is non-convex, the PARAFAC2 model suffers from local minima. In this paper, we investigated the local minima issue in the PARAFAC2 decomposition. We observed that the decomposed loading matrices corresponding to local minima and global minima are in general very different even though the loss function values can be very similar. We also observed that overfitting always led to higher fraction of local minima. For coping with local minima in PARAFAC2, we investigated the effect of non-negativity constraints on avoiding local minima, proposed some PARAFAC2 algorithms with optimized initializations, and implemented a simple local minima avoidance procedure in the general PARAFAC2 algorithm. These remedies are evaluated and illustrated by the use of different types of datasets such as simulated data, GC-MS data and EEG data. It is finally concluded that imposing non-negativity constraints on the PARAFAC2 model decreased the fraction of local minima significantly, using the proposed optimized initialization PARAFAC2 algorithm decreased the average fraction of local minima from the highest (84%) to the lowest (0%), and using the proposed local minima avoidance PARAFAC2 algorithm eliminated all the local minima in our case. In case of modeling data with high complexity, it will be promising to use all the remedies at the same time in order to avoid local minima PARAFAC2 models successfully. To the best of our knowledge, this is the first paper that investigate the local minima issue in the context of PARAFAC2 for irregular tensor.

Resource usage of traditional quantum computing simulation techniques scale exponentially fast either with the number of qubits or the depth of the circuit. Simulations of medium-sized circuits are then intractable but new methods have appeared on the literature with better scaling than traditional methods. Such is the case of tensor networks, whose main advantage is that they require ex- ponentially less space and can trade off between fidelity and memory usage. In this work, I develop a distributed quantum computing simulator based on tensor network methods capable of exactly simulating quantum circuits.

Магістерська дисертація:91 с., 15 рис., 18 табл., 1 додаток, 17 джерел. Тема магістерської дисертації «Нейромережевий вибір найкращої моделі прогнозування». Мета роботи — розробка програми для вибору найкращої моделі серед запропонованих нейромереж Об’єкт дослідження - нейронні мережі, їх можливості та перспективи у сфері прогнозування. Предмет дослідження - моделі та методи застосування нейронних мереж для задач прогнозування, шляхи покращення існуючих методів та систем прогнозування. Для досягнення мети були поставлення наступні задачі: 1. Огляд предметної області та аналіз існуючих рішень, архітектур нейромереж; 2. Розробка нових підходів до прогнозування на основі використання елементів штучного інтелекту; 3. Розробка програмного комплексу, що забезпечуватиме просте використання існуючих та розроблених методів для вирішення задачі прогнозування часових рядів. Програмний продукт реалізовано з використанням мови програмування Python який надає широкий спектр бібліотек для обробки та аналізу даних., Master's thesis:91 pp., 15 fig., 18 tabl., 1 application, 17 sources. The theme of my master's thesis is «Neural network selection of the best forecasting model». The purpose of the work is to develop a program for choosing the best model among the proposed neural networks The object of the work is neural networks, their capabilities and prospects in the field of forecasting. The subject of the work - models and methods of application of neural networks for forecasting tasks, ways to improve existing methods and forecasting systems. To achieve this goal, the following tasks were set: 1. Review of the subject area and analysis of existing solutions, architectures neural networks; 2. Development of new approaches to forecasting based on use elements of artificial intelligence; 3. Development of a software package that will provide ease of use existing and developed methods for solving the problem of forecasting time series. The software product is implemented using the Python programming language which provides a wide range of libraries for data processing and analysis.

The recent emergence of complex datasets in various disciplines presents a pressing need to devise regression models that have tensors either as a response or as a covariate, often under assumptions of sparsity in the corresponding tensor-valued coefficients. Models that involve tensors often require special treatment in a modeling setting due to their potentially large structures and general assumptions of sparsity in regard to associations with covariates. Importantly, scenarios with small sample sizes benefit from Bayesian methods that allow for flexible model conditions while also rigorously defining the uncertainty in any conclusions drawn from a model. We begin with general introductions to Bayesian analysis, neuroimaging, and tensor notations in the first chapter. The goal in these short overviews is not to provide a comprehensive background, but to inform the casual reader about key concepts that will be referenced throughout this dissertation. Afterwards, we proceed through new methods in Bayesian modeling of tensor-valued variables, covering three different analysis scenarios. The goal in the second chapter is to develop a Bayesian tensor response regression in order to identify contiguous spatial regions that are associated with a given covariate. The method is then applied to detecting neuronal activation in functional magnetic resonance imaging (fMRI) experiments in the presence of tensor-valued brain images and a scalar predictor for a single subject. We propose to regress responses from all cells (called voxels in brain activation studies) together as a tensor response on scalar predictors, accounting for the structural information inherent in the tensor response. To estimate model parameters with proper cell specific shrinkage, we propose a novel \emph{multiway stick breaking shrinkage prior} distribution on tensor structured regression coefficients, enabling identification of cells which are related to the predictors. The major novelty of this chapt

La primera parte del trabajo describe la noción de tensor, prestando especial atención al enfoque libre de coordenadas, sin dejar no obstante de lado su tratamiento clásico como datos de mediciones con respecto de un sistema de referencia (enfoque con coordenadas). Se incluye además un estudio detallado de los campos tensoriales, como extensión del concepto de tensor, tratando en especial los tensores y campos tensoriales métricos dada la importancia de los resultados que derivan de ellos. La segunda parte aborda el concepto de dato tensorial, desde el punto de vista de la generalización n-dimensional de las nociones de escalar, vector y matriz. A partir de esta generalización del enfoque con coordenadas, se tratarán sus operaciones prácticas actuales, junto con las descomposiciones tensoriales más relevantes en la actualidad. Estas descomposiciones se utilizan para generar algoritmos aplicables en numerosas áreas actuales, como en el estudio de señales, visión computacional, neurociencia, teoría de grafos, minería de datos y aprendizaje automático, entre otros. El trabajo trata de unificar y contrastar la terminología clásica y la práctica actual., Grado en Matemáticas

The recent emergence of complex datasets in various disciplines presents a pressing need to devise regression models that have tensors either as a response or as a covariate, often under assumptions of sparsity in the corresponding tensor-valued coefficients. Models that involve tensors often require special treatment in a modeling setting due to their potentially large structures and general assumptions of sparsity in regard to associations with covariates. Importantly, scenarios with small sample sizes benefit from Bayesian methods that allow for flexible model conditions while also rigorously defining the uncertainty in any conclusions drawn from a model. We begin with general introductions to Bayesian analysis, neuroimaging, and tensor notations in the first chapter. The goal in these short overviews is not to provide a comprehensive background, but to inform the casual reader about key concepts that will be referenced throughout this dissertation. Afterwards, we proceed through new methods in Bayesian modeling of tensor-valued variables, covering three different analysis scenarios. The goal in the second chapter is to develop a Bayesian tensor response regression in order to identify contiguous spatial regions that are associated with a given covariate. The method is then applied to detecting neuronal activation in functional magnetic resonance imaging (fMRI) experiments in the presence of tensor-valued brain images and a scalar predictor for a single subject. We propose to regress responses from all cells (called voxels in brain activation studies) together as a tensor response on scalar predictors, accounting for the structural information inherent in the tensor response. To estimate model parameters with proper cell specific shrinkage, we propose a novel \emph{multiway stick breaking shrinkage prior} distribution on tensor structured regression coefficients, enabling identification of cells which are related to the predictors. The major novelty of this chapt

La primera parte del trabajo describe la noción de tensor, prestando especial atención al enfoque libre de coordenadas, sin dejar no obstante de lado su tratamiento clásico como datos de mediciones con respecto de un sistema de referencia (enfoque con coordenadas). Se incluye además un estudio detallado de los campos tensoriales, como extensión del concepto de tensor, tratando en especial los tensores y campos tensoriales métricos dada la importancia de los resultados que derivan de ellos. La segunda parte aborda el concepto de dato tensorial, desde el punto de vista de la generalización n-dimensional de las nociones de escalar, vector y matriz. A partir de esta generalización del enfoque con coordenadas, se tratarán sus operaciones prácticas actuales, junto con las descomposiciones tensoriales más relevantes en la actualidad. Estas descomposiciones se utilizan para generar algoritmos aplicables en numerosas áreas actuales, como en el estudio de señales, visión computacional, neurociencia, teoría de grafos, minería de datos y aprendizaje automático, entre otros. El trabajo trata de unificar y contrastar la terminología clásica y la práctica actual., Grado en Matemáticas