As doenças cardiovasculares (DCV) representam a principal causa de mortalidade e de incapacidade, em ambos os gêneros, no Brasil e no mundo. O padrão de consumo alimentar está tanto positiva como negativamente associado aos principais fatores de risco para DCV, entre eles diabetes, hipertensão, obesidade e hipertrigliceridemia, todos componentes da síndrome metabólica. Este estudo tem como objetivos identificar os padrões alimentares em indivíduos com DCV, considerando a densidade de energia, a gordura saturada, a fibra, o sódio e o potássio consumidos, e investigar sua associação com fatores de risco de DCV e síndrome metabólica. Trata-se de um estudo transversal. Foram utilizados dados do estudo DICA Br. A amostra foi composta de indivíduos com DCV, com idade superior a 45 anos, de todas as regiões brasileiras. O consumo alimentar foi obtido por recordatório alimentar de 24h e os padrões alimentares obtidos pela regressão por posto reduzido (RPR). Para a RPR, utilizaram-se 28 grupos alimentares como preditores e como variáveis respostas componentes dietéticos. O teste de Mann Whitney foi utilizado para testar as diferenças entre as médias dos escores. Foram obtidos dados de 1.047 participantes; 95% apresentavam doença arterial coronariana; em sua maioria, eram idosos, da classe econômica C1 e C2 e estudaram até o ensino médio. A prevalência de síndrome metabólica foi de 58%. Foram extraídos dois padrões alimentares. O primeiro foi marcado pelo maior consumo de fibra alimentar e potássio, composto por arroz e feijão, frutas e sucos naturais com ou sem açúcar, legumes, carne bovina ou processada, verduras, raízes e tubérculos. O segundo padrão caracterizou-se pelo consumo de gordura saturada e maior densidade energética, representado por panificados salgados, gorduras, carne bovina e processada, doces caseiros, pizza, salgadinhos de pacote ou festa, sanduíche e alimento salgado pronto para consumo. Houve associação significativa entre o padrão alimentar 1 com medida da circunferência da cintura e nível de HDL adequados e com o padrão 2 e HDL adequado. A adoção do padrão alimentar 1 pode estar associada à proteção contra alguns dos componentes da síndrome metabólica. / Cardiovascular diseases (CVD) are the leading cause of mortality and disability in both genders in Brazil and worldwide. The dietary pattern is at the same time positive and negatively associated with the main risk factors for CVD, including diabetes, hypertension, obesity and hypertriglyceridemia, all components of the metabolic syndrome. This study aims to identify dietary patterns in individuals with CVD, considering the energy density, and the amount of saturated fatty acid, fiber, sodium and potassium of the diet, and to investigate its association with CVD risk factors and metabolic syndrome. This is a cross-sectional study, data were used from \"DICA Br\" study. The sample consisted of individuals with CVD, over 45 years old, residents from all Brazilian regions. Food consumption was obtained by one 24-hours diet recall and dietary patterns by reduced rank regression (RRR). In the RRR, 28 food groups were included as predictors and dietary components was chosen as the response variable. The Mann-Whitney test was used to test the differences between the factors scores\' means. Data of 1047 participants were analyzed. 95% have coronary artery disease, most are elderly, economical class most observed are C1 and C2. Also, most of them and studied up to high school. The prevalence of metabolic syndrome was 58%. Two dietary patterns were extracted: the first one is higher in dietary fiber and potassium, which is composed by rice, beans, fruits and natural juices with or without sugar, vegetables, beef or processed meat, roots and tubers. The second pattern is higher in saturated fatty acid and energy density, represented by breads, fats, and processed meat, homemade pastries, pizza, snacks or party package, sandwich and salty food ready for consumption. There was a significant association between dietary pattern 1 and low waist circumference and adequate high density cholesterol blood concentration. There was a significant association between dietary pattern 2 and adequate high density cholesterol blood concentration. We suggest that the adoption of the dietary pattern 1 may be associated with protection against some of the components of metabolic syndrome.

Ce manuscrit regroupe différents travaux explorant les interactions entre les tenseurs et l'apprentissage automatique. Le premier chapitre est consacré à l'extension des modèles de séries reconnaissables de chaînes et d'arbres aux graphes. Nous y montrons que les modèles d'automates pondérés de chaînes et d'arbres peuvent être interprétés d'une manière simple et unifiée à l'aide de réseaux de tenseurs, et que cette interprétation s'étend naturellement aux graphes ; nous étudions certaines propriétés de ce modèle et présentons des résultats préliminaires sur leur apprentissage. Le second chapitre porte sur la minimisation approximée d'automates pondérés d'arbres et propose une approche théoriquement fondée à la problématique suivante : étant donné un automate pondéré d'arbres à n états, comment trouver un automate à m<n états calculant une fonction proche de l'originale. Le troisième chapitre traite de la régression de faible rang pour sorties à structure tensorielle. Nous y proposons un algorithme d'apprentissage rapide et efficace pour traiter un problème de régression dans lequel les sorties des tenseurs. Nous montrons que l'algorithme proposé est un algorithme d'approximation pour ce problème NP-difficile et nous donnons une analyse théorique de ses propriétés statistiques et de généralisation. Enfin, le quatrième chapitre introduit le modèle de mélanges algébriques de distributions. Ce modèle considère des combinaisons affines de distributions (où les coefficients somment à un mais ne sont pas nécessairement positifs). Nous proposons une approche pour l'apprentissage de mélanges algébriques qui étend la méthode tensorielle des moments introduite récemment. . / This thesis tackles several problems exploring connections between tensors and machine learning. In the first chapter, we propose an extension of the classical notion of recognizable function on strings and trees to graphs. We first show that the computations of weighted automata on strings and trees can be interpreted in a natural and unifying way using tensor networks, which naturally leads us to define a computational model on graphs: graph weighted models; we then study fundamental properties of this model and present preliminary learning results. The second chapter tackles a model reduction problem for weighted tree automata. We propose a principled approach to the following problem: given a weighted tree automaton with n states, how can we find an automaton with m<n states that is a good approximation of the original one? In the third chapter, we consider a problem of low rank regression for tensor structured outputs. We design a fast and efficient algorithm to address a regression task where the outputs are tensors. We show that this algorithm generalizes the reduced rank regression method and that it offers good approximation, statistical and generalization guarantees. Lastly in the fourth chapter, we introduce the algebraic mixture model. This model considers affine combinations of probability distributions (where the weights sum to one but may be negative). We extend the recently proposed tensor method of moments to algebraic mixtures, which allows us in particular to design a learning algorithm for algebraic mixtures of spherical Gaussian distributions.

Multivariate linear regression methods are widely used statistical tools in data analysis, and were developed when some response variables are studied simultaneously, in which our aim is to study the relationship between predictor variables and response variables through the regression coefficient matrix. The rapid improvements of information technology have brought us a large number of large-scale data, but also brought us great challenges in data processing. When dealing with high dimensional data, the classical least squares estimation is not applicable in multivariate linear regression analysis. In recent years, some approaches have been developed to deal with high-dimensional data problems, among which dimension reduction is one of the main approaches. In some literature, random projection methods were used to reduce dimension in large datasets. In Chapter 2, a new random projection method, with low-rank matrix approximation, is proposed to reduce the dimension of the parameter space in high-dimensional multivariate linear regression model. Some statistical properties of the proposed method are studied and explicit expressions are then derived for the accuracy loss of the method with Gaussian random projection and orthogonal random projection. These expressions are precise rather than being bounds up to constants. In multivariate regression analysis, reduced rank regression is also a dimension reduction method, which has become an important tool for achieving dimension reduction goals due to its simplicity, computational efficiency and good predictive performance. In practical situations, however, the performance of the reduced rank estimator is not satisfactory when the predictor variables are highly correlated or the ratio of signal to noise is small. To overcome this problem, in Chapter 3, we incorporate matrix projections into reduced rank regression method, and then develop reduced rank regression estimators based on random projection and orthogonal projection in high-dimensional multivariate linear regression models. We also propose a consistent estimator of the rank of the coefficient matrix and achieve prediction performance bounds for the proposed estimators based on mean squared errors. Envelope technology is also a popular method in recent years to reduce estimative and predictive variations in multivariate regression, including a class of methods to improve the efficiency without changing the traditional objectives. Variable selection is the process of selecting a subset of relevant features variables for use in model construction. The purpose of using this technology is to avoid the curse of dimensionality, simplify models to make them easier to interpret, shorten training time and reduce overfitting. In Chapter 4, we combine envelope models and a group variable selection method to propose an envelope-based sparse reduced rank regression estimator in high-dimensional multivariate linear regression models, and then establish its consistency, asymptotic normality and oracle property. Tensor data are in frequent use today in a variety of fields in science and engineering. Processing tensor data is a practical but challenging problem. Recently, the prevalence of tensor data has resulted in several envelope tensor versions. In Chapter 5, we incorporate envelope technique into tensor regression analysis and propose a partial tensor envelope model, which leads to a parsimonious version for tensor response regression when some predictors are of special interest, and then consistency and asymptotic normality of the coefficient estimators are proved. The proposed method achieves significant gains in efficiency compared to the standard tensor response regression model in terms of the estimation of the coefficients for the selected predictors. Finally, in Chapter 6, we summarize the work carried out in the thesis, and then suggest some problems of further research interest. / Dissertation / Doctor of Philosophy (PhD)

As IC technologies scale to the nanometer regime, efficient and accurate modeling and analysis of VLSI systems with billions of transistors and interconnects becomes increasingly critical and difficult. VLSI systems impacted by the increasingly high dimensional process-voltage-temperature (PVT) variations demand much more modeling and analysis efforts than ever before, while the analysis of large scale on-chip interconnects that requires solving tens of millions of unknowns imposes great challenges in computer aided design areas. This dissertation presents new methodologies for addressing the above two important challenging issues for large scale on-chip interconnect modeling and analysis: In the past, the standard statistical circuit modeling techniques usually employ principal component analysis (PCA) and its variants to reduce the parameter dimensionality. Although widely adopted, these techniques can be very limited since parameter dimension reduction is achieved by merely considering the statistical distributions of the controlling parameters but neglecting the important correspondence between these parameters and the circuit performances (responses) under modeling. This dissertation presents a variety of performance-oriented parameter dimension reduction methods that can lead to more than one order of magnitude parameter reduction for a variety of VLSI circuit modeling and analysis problems. The sheer size of present day power/ground distribution networks makes their analysis and verification tasks extremely runtime and memory inefficient, and at the same time, limits the extent to which these networks can be optimized. Given today?s commodity graphics processing units (GPUs) that can deliver more than 500 GFlops (Flops: floating point operations per second). computing power and 100GB/s memory bandwidth, which are more than 10X greater than offered by modern day general-purpose quad-core microprocessors, it is very desirable to convert the impressive GPU computing power to usable design automation tools for VLSI verification. In this dissertation, for the first time, we show how to exploit recent massively parallel single-instruction multiple-thread (SIMT) based graphics processing unit (GPU) platforms to tackle power grid analysis with very promising performance. Our GPU based network analyzer is capable of solving tens of millions of power grid nodes in just a few seconds. Additionally, with the above GPU based simulation framework, more challenging three-dimensional full-chip thermal analysis can be solved in a much more efficient way than ever before.

In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors. The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is \[ \min_{\rank(X) = k} \det (B - X A)(B - X A)\tp, \] where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix. Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations. The remaining two papers discuss computational methods for the best multilinear rank approximation problem \[ \min_{\cB} \| \cA - \cB\| \] where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product \[ \cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu}, \] where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.