All procedures in the literature for decomposing symmetric second rank (stress) tensor and symmetric fourth rank (elastic constant) tensor are elaborated and compared which have many engineering and scientific applications for anisotropic materials. The decomposition methods for symmetric second rank tensors are orthonormal tensor basis method, complex variable representation and spectral method. For symmetric fourth rank (elastic constant) tensor, there are four mainly decomposition methods namely as, orthonormal tensor basis, irreducible, harmonic decomposition and spectral. Those are applied to anisotropic materials possessing various symmetry classes which are isotropic, cubic, transversely isotropic, tetragonal, trigonal and orthorhombic. For isotropic materials, an expression for the elastic constant tensor different than the traditionally known form is given. Some misprints found in the literature are corrected. For comparison purposes, numerical examples of each decomposition process are presented for the materials possessing different symmetry classes. Some applications of these decomposition methods are given. Besides, norm and norm ratio concepts are introduced to measure and compare the anisotropy degree for various materials with the same or di&curren / erent symmetries. For these materials,norm and norm ratios are calculated. It is suggested that the norm of a tensor may be used as a criterion for comparing the overall e&curren / ect of the properties of anisotropic materials and the norm ratios may be used as a criterion to represent the anisotropy degree of the properties of materials. Finally, comparison of all methods are done in order to determine similarities and differences between them. As a result of this comparison process, it is proposed that the spectral method is a non-linear decomposition method which yields non-linear orthogonal decomposed parts. For symmetric second rank and fourth rank tensors, this case is a significant innovation in decomposition procedures in the literature.

La disponibilité et la persistance à l'échelle locale des métaux lourds pourraient être critiques notamment pour l'usage futur des zones agricoles ou urbaines, au droit desquelles de nombreux sites industriels se sont installés dans le passé. La gestion de ces situations environnementales complexes nécessitent le développement de nouvelles méthodes d'analyse peu invasives (capteurs environnementaux), comme celles utilisant des biosenseurs bactériens, afin d'identifier et d'évaluer directement l'effet biologique et la disponibilité chimique des métaux. Ainsi dans ce travail de thèse, nous avons cherché à identifier, à l'aide d'outils mathématiques de l'algèbre multilinéaire, les réponses de senseurs bactériens fluorescents dans des conditions environnementales variées, qu'il s'agisse d'un stress engendré par la présence à forte dose d'un métal ou d'une carence nutritive engendrée par son absence. Cette identification est fondée sur l'analyse quantitative à l'échelle d'une population bactérienne de signaux multidimensionnels. Elle repose en particulier sur (i) l'acquisition de données spectrales (fluorescence) multi-variées sur des suspensions de biosenseurs multicolores interagissant avec des métaux et sur (ii) le développement d'algorithme de décomposition tensoriels. Les méthodes proposées, développées et utilisées dans ce travail s'efforcent d'identifier « sans a priori» a minima, la réponse fonctionnelle de biosenseurs sous différentes conditions environnementales, par des méthodes de décomposition de tenseurs sous contraintes des signaux spectraux observables. Elles tirent parti de la variabilité des réponses systémiques et permettent de déterminer les sources élémentaires identifiant le système et leur comportement en fonction des paramètres extérieurs. Elles sont inspirées des méthodes CP et PARALIND . L'avantage de ce type d'approche, par rapport aux approches classiques, est l'identification unique des réponses des biosenseurs sous de faibles contraintes. Le travail a consisté à développer des algorithmes efficaces de séparations de sources pour les signaux fluorescents émis par des senseurs bactériens, garantissant la séparabilité des sources fluorescentes et l'unicité de la décomposition. Le point original de la thèse est la prise en compte des contraintes liées à la physique des phénomènes analysés telles que (i) la parcimonie des coefficients de mélange ou la positivité des signaux source, afin de réduire au maximum l'usage d'a priori ou (ii) la détermination non empirique de l'ordre de la décomposition (nombre de sources). Cette posture a permis aussi d'améliorer l'identification en optimisant les mesures physiques par l'utilisation de spectres synchrones ou en apportant une diversité suffisante aux plans d'expériences. L'usage des spectres synchrones s'est avéré déterminant à la fois pour améliorer la séparation des sources de fluorescence, mais aussi pour augmenter le rapport signal sur bruit des biosenseurs les plus faibles. Cette méthode d'analyse spectrale originale permet d'élargir fortement la gamme chromatique des biosenseurs fluorescents multicolores utilisables simultanément. Enfin, une nouvelle méthode d'estimation de la concentration de polluants métalliques présents dans un échantillon à partir de la réponse spectrale d'un mélange de biosenseurs non-spécifiques a été développée / Availability and persistence of heavy metals could be critical for future use of agricultural or urban areas, on which many industrial sites have installed in the past. The management of these complex environmental situations requiring the development of new analytical methods minimally invasive, such as bacterial biosensors, to identify and directly assess the biological effects and the chemical availability of metals. The aims of this thesis was to identify the responses of fluorescent bacterial sensors various environmental conditions, using mathematical tools of algebra multi-linear, whether stress caused by the presence of high dose of a metal or a nutrient deficiency caused by his absence. This identification is based on quantitative analysis of multidimensional signals at the bacterial population-scale. It is based in particular on (i) the acquisition of multivariate spectral data on suspensions of multicolored biosensors interacting with metals and (ii) the development of algorithms for tensor decomposition. The proposed methods, developed and used in this study attempt to identify functional response of biosensors without \textsl{a priori} by decomposition of tensor containing the spectral signals. These methods take advantage of the variability of systemic responses and allow to determine the basic sources identifying the system and their behavior to external factors. They are inspired by the CP and PARALIND methods. The advantage of this approach, compared to conventional approaches, is the unique identification of the responses of biosensors at low constraints. The work was to develop efficient algorithms for the source separation of fluorescent signals emitted by bacterial sensors, ensuring the sources separability and the uniqueness of the decomposition. The original point of this thesis is the consideration of the physical constraints of analyzed phenomena such as (i) the sparsity of mixing coefficients or positivity of sources signals in order to minimize the use of a priori or (ii) the non-empirical determination of the order of decomposition (number of sources).This posture has also improved the identification optimizing physical measurements by the use of synchronous spectra or providing sufficient diversity in design of experiments. The use of synchronous spectra proved crucial both to improve the separation of fluorescent sources, but also to increase the signal to noise ratio of the lowest biosensors. This original method of spectral analysis can greatly expand the color range of multicolored fluorescent biosensors used simultaneously. Finally, a new method of estimating the concentration of metal pollutants present in a sample from the spectral response of a mixture of non-specific biosensor was developed

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.

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This thesis introduces a multilinear algebraic framework for computer graphics, computer vision, and machine learning, particularly for the fundamental purposes of image synthesis, analysis, and recognition. Natural images result from the multifactor interaction between the imaging process, the scene illumination, and the scene geometry. We assert that a principled mathematical approach to disentangling and explicitly representing these causal factors, which are essential to image formation, is through numerical multilinear algebra, the algebra of higher-order tensors. Our new image modeling framework is based on(i) a multilinear generalization of principal components analysis (PCA), (ii) a novel multilinear generalization of independent components analysis (ICA), and (iii) a multilinear projection for use in recognition that maps images to the multiple causal factor spaces associated with their formation. Multilinear PCA employs a tensor extension of the conventional matrix singular value decomposition (SVD), known as the M-mode SVD, while our multilinear ICA method involves an analogous M-mode ICA algorithm. As applications of our tensor framework, we tackle important problems in computer graphics, computer vision, and pattern recognition; in particular, (i) image-based rendering, specifically introducing the multilinear synthesis of images of textured surfaces under varying view and illumination conditions, a new technique that we call ``TensorTextures'', as well as (ii) the multilinear analysis and recognition of facial images under variable face shape, view, and illumination conditions, a new technique that we call ``TensorFaces''. In developing these applications, we introduce a multilinear image-based rendering algorithm and a multilinear appearance-based recognition algorithm. As a final, non-image-based application of our framework, we consider the analysis, synthesis and recognition of human motion data using multilinear methods, introducing a new technique that we call ``Human Motion Signatures''.

This thesis introduces a multilinear algebraic framework for computer graphics, computer vision, and machine learning, particularly for the fundamental purposes of image synthesis, analysis, and recognition. Natural images result from the multifactor interaction between the imaging process, the scene illumination, and the scene geometry. We assert that a principled mathematical approach to disentangling and explicitly representing these causal factors, which are essential to image formation, is through numerical multilinear algebra, the algebra of higher-order tensors. Our new image modeling framework is based on(i) a multilinear generalization of principal components analysis (PCA), (ii) a novel multilinear generalization of independent components analysis (ICA), and (iii) a multilinear projection for use in recognition that maps images to the multiple causal factor spaces associated with their formation. Multilinear PCA employs a tensor extension of the conventional matrix singular value decomposition (SVD), known as the M-mode SVD, while our multilinear ICA method involves an analogous M-mode ICA algorithm. As applications of our tensor framework, we tackle important problems in computer graphics, computer vision, and pattern recognition; in particular, (i) image-based rendering, specifically introducing the multilinear synthesis of images of textured surfaces under varying view and illumination conditions, a new technique that we call ``TensorTextures'', as well as (ii) the multilinear analysis and recognition of facial images under variable face shape, view, and illumination conditions, a new technique that we call ``TensorFaces''. In developing these applications, we introduce a multilinear image-based rendering algorithm and a multilinear appearance-based recognition algorithm. As a final, non-image-based application of our framework, we consider the analysis, synthesis and recognition of human motion data using multilinear methods, introducing a new technique that we call ``Human Motion Signatures''.