Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor-based regression models and applications

Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2017-2018Avec l’avancement des technologies modernes, les tenseurs d’ordre élevé sont assez répandus et abondent dans un large éventail d’applications telles que la neuroscience informatique, la vision par ordinateur, le traitement du signal et ainsi de suite. La principale raison pour laquelle les méthodes de régression classiques ne parviennent pas à traiter de façon appropriée des tenseurs d’ordre élevé est due au fait que ces données contiennent des informations structurelles multi-voies qui ne peuvent pas être capturées directement par les modèles conventionnels de régression vectorielle ou matricielle. En outre, la très grande dimensionnalité de l’entrée tensorielle produit une énorme quantité de paramètres, ce qui rompt les garanties théoriques des approches de régression classique. De plus, les modèles classiques de régression se sont avérés limités en termes de difficulté d’interprétation, de sensibilité au bruit et d’absence d’unicité. Pour faire face à ces défis, nous étudions une nouvelle classe de modèles de régression, appelés modèles de régression tensor-variable, où les prédicteurs indépendants et (ou) les réponses dépendantes prennent la forme de représentations tensorielles d’ordre élevé. Nous les appliquons également dans de nombreuses applications du monde réel pour vérifier leur efficacité et leur efficacité.With the advancement of modern technologies, high-order tensors are quite widespread and abound in a broad range of applications such as computational neuroscience, computer vision, signal processing and so on. The primary reason that classical regression methods fail to appropriately handle high-order tensors is due to the fact that those data contain multiway structural information which cannot be directly captured by the conventional vector-based or matrix-based regression models, causing substantial information loss during the regression. Furthermore, the ultrahigh dimensionality of tensorial input produces huge amount of parameters, which breaks the theoretical guarantees of classical regression approaches. Additionally, the classical regression models have also been shown to be limited in terms of difficulty of interpretation, sensitivity to noise and absence of uniqueness. To deal with these challenges, we investigate a novel class of regression models, called tensorvariate regression models, where the independent predictors and (or) dependent responses take the form of high-order tensorial representations. We also apply them in numerous real-world applications to verify their efficiency and effectiveness. Concretely, we first introduce hierarchical Tucker tensor regression, a generalized linear tensor regression model that is able to handle potentially much higher order tensor input. Then, we work on online local Gaussian process for tensor-variate regression, an efficient nonlinear GPbased approach that can process large data sets at constant time in a sequential way. Next, we present a computationally efficient online tensor regression algorithm with general tensorial input and output, called incremental higher-order partial least squares, for the setting of infinite time-dependent tensor streams. Thereafter, we propose a super-fast sequential tensor regression framework for general tensor sequences, namely recursive higher-order partial least squares, which addresses issues of limited storage space and fast processing time allowed by dynamic environments. Finally, we introduce kernel-based multiblock tensor partial least squares, a new generalized nonlinear framework that is capable of predicting a set of tensor blocks by merging a set of tensor blocks from different sources with a boosted predictive power

Score Function Features for Discriminative Learning: Matrix and Tensor Framework

Feature learning forms the cornerstone for tackling challenging learning problems in domains such as speech, computer vision and natural language processing. In this paper, we consider a novel class of matrix and tensor-valued features, which can be pre-trained using unlabeled samples. We present efficient algorithms for extracting discriminative information, given these pre-trained features and labeled samples for any related task. Our class of features are based on higher-order score functions, which capture local variations in the probability density function of the input. We establish a theoretical framework to characterize the nature of discriminative information that can be extracted from score-function features, when used in conjunction with labeled samples. We employ efficient spectral decomposition algorithms (on matrices and tensors) for extracting discriminative components. The advantage of employing tensor-valued features is that we can extract richer discriminative information in the form of an overcomplete representations. Thus, we present a novel framework for employing generative models of the input for discriminative learning.Comment: 29 page

Experimental Design for Sensitivity Analysis, Optimization and Validation of Simulation Models

This chapter gives a survey on the use of statistical designs for what-if analysis in simula- tion, including sensitivity analysis, optimization, and validation/verification. Sensitivity analysis is divided into two phases. The first phase is a pilot stage, which consists of screening or searching for the important factors among (say) hundreds of potentially important factors. A novel screening technique is presented, namely sequential bifurcation. The second phase uses regression analysis to approximate the input/output transformation that is implied by the simulation model; the resulting regression model is also known as a metamodel or a response surface. Regression analysis gives better results when the simu- lation experiment is well designed, using either classical statistical designs (such as frac- tional factorials) or optimal designs (such as pioneered by Fedorov, Kiefer, and Wolfo- witz). To optimize the simulated system, the analysts may apply Response Surface Metho- dology (RSM); RSM combines regression analysis, statistical designs, and steepest-ascent hill-climbing. To validate a simulation model, again regression analysis and statistical designs may be applied. Several numerical examples and case-studies illustrate how statisti- cal techniques can reduce the ad hoc character of simulation; that is, these statistical techniques can make simulation studies give more general results, in less time. Appendix 1 summarizes confidence intervals for expected values, proportions, and quantiles, in termi- nating and steady-state simulations. Appendix 2 gives details on four variance reduction techniques, namely common pseudorandom numbers, antithetic numbers, control variates or regression sampling, and importance sampling. Appendix 3 describes jackknifing, which may give robust confidence intervals.least squares;distribution-free;non-parametric;stopping rule;run-length;Von Neumann;median;seed;likelihood ratio

Self-similar prior and wavelet bases for hidden incompressible turbulent motion

This work is concerned with the ill-posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence-free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self-similar prior characterizes accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergent-free fBm on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence-free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposition involves correlated wavelet coefficients, we are able to handle this dependence in practice. Based on these two wavelet decompositions, we finally provide effective and efficient algorithms to approach the maximum a posteriori. An intensive numerical evaluation proves the relevance of the proposed wavelet-based self-similar priors.Comment: SIAM Journal on Imaging Sciences, 201

Learning Theory and Approximation

The main goal of this workshop – the third one of this type at the MFO – has been to blend mathematical results from statistical learning theory and approximation theory to strengthen both disciplines and use synergistic effects to work on current research questions. Learning theory aims at modeling unknown function relations and data structures from samples in an automatic manner. Approximation theory is naturally used for the advancement and closely connected to the further development of learning theory, in particular for the exploration of new useful algorithms, and for the theoretical understanding of existing methods. Conversely, the study of learning theory also gives rise to interesting theoretical problems for approximation theory such as the approximation and sparse representation of functions or the construction of rich kernel reproducing Hilbert spaces on general metric spaces. This workshop has concentrated on the following recent topics: Pitchfork bifurcation of dynamical systems arising from mathematical foundations of cell development; regularized kernel based learning in the Big Data situation; deep learning; convergence rates of learning and online learning algorithms; numerical refinement algorithms to learning; statistical robustness of regularized kernel based learning