In this paper, the approach of utilizing the core tensor from the Tucker decomposition, in place of theuncompressed tensor, for nding a valid tensor rank for the PARAFAC decomposition is considered.Validity of the proposed method is investigated in terms of error and time consumption. As thesolutions of the PARAFAC decomposition are unique, stability of the solutions through split-halfanalysis is investigated. Simulated and real data are considered. Although, no general validity ofthe method could be observed, the results for some datasets look promising with 10% compressionin all modes. It is also shown that increased compression does not necessarily imply less timeconsumption.

Cette thèse est consacrée à l’étude des rangs matriciels et tensoriels des données multidimensionnelles, et au développement de méthodes d’estimation de ces rangs dans le cadre de la transformée en ondelettes. Pour cette étude, nous avons eu re-cours à la décomposition en paquets d’ondelettes et à l’algèbre multilinéaire. Une méthode d’optimisation stochastique bio-inspirée a été adaptée, avec pour objectif final de supprimer le bruit dans des images multidimensionnelles. Pour cela nous avons estimé les différentes valeurs des dimensions du sous-espace de tenseur pour tous les modes des coefficients des paquets d’ondelettes. Nous avons appliqué les méthodes de débruitage proposées à diverses images multidimensionnelles : images RGB, images multispectrales extraites d’images hyperspectrales de pièces métalliques, images par fluorescence des plantes, et images RX multispectrales. Finalement, une étude comparative a été réalisée avec trois principaux types d’algorithmes : d’une part, la méthode de Perona-Malik basée sur la diffusion ; deuxièmement, la troncature de HOSVD (Higher-Order Singular Value Decomposition) et MWF (Multiway Wiener Filtering) et troisièmement, un procédé basé sur la dé- composition en paquets d’ondelettes et MWF (Multiway Wiener Filtering), où les dimensions du sous-espace de signal sont estimées par un critère statistique plutôt que par une méthode d’optimisation. Les résultats sont prometteurs en termes de débruitage en réalité terrain. En définitive, nous aboutissons à un gain de temps avantageux durant le traitement des images hyperspectrales. / This thesis is devoted to study matrix and tensor ranks of multidimensional signalsand to the development of methods for estimating these ranks in the frameworkof the wavelet transform. For this study, we used the wavelet packet decompositionand the multilinear algebra. A bio-inspired stochastic optimization methodhas been adapted, with the ultimate objective of suppressing noise in multidimensionalimages. In order to ensure this, we have estimated the different values ofthe dimensions of the tensor subspace for all the modes of the coefficients of thewavelet packets.We have applied the proposed denoising methods to various multidimensionalimages: RGB images, multispectral images extracted from hyperspectralimages of metal parts, plant fluorescence images, and multispectral RX images.Finally, a comparative study was carried out with three main types of algorithms: onthe one hand, the Perona-Malik method based on diffusion; Second, the truncationof HOSVD and MWF, and thirdly, a method based on wavelet packet decompositionand MWF, where the dimensions of the signal subspace are estimated by a statisticalcriterion rather than by an optimization method. The results are promising in termsof denoising in grund truth. Ultimately, we achieve an advantageous time savingduring the acquisition of hyperspectral images.

On s'intéresse dans cette thèse à la complexité bilinéaire de la multiplication dans toute extension de degré $n$ d'un corps $F_{q}$ à $q$ éléments, qui est étroitement liée au rang de tenseur de la multiplication dans $F_{q^n}$. L'algorithme de type évaluation-interpolation, introduit par D.V et G.V Chudnovsky en 1987, est à la base des techniques algorithmiques fournissant actuellement les meilleures bornes uniformes et asymptotiques de la complexité bilinéaire. Pour obtenir ces meilleures bornes, la stratégie adoptée jusqu'à présent consistait à fixer le degré des places en augmentant le genre du corps de fonctions algébriques.Néanmoins, l'étude de la construction effective associée à ce type de stratégie fut jusqu'à présent négligée en raison d'un obstacle, lié à la construction du point de degré $n$, relevé par Shparlinski, Tsfasman et Vladut en 1992.On présente dans cette thèse une nouvelle stratégie qui consiste à fixer le genre du corps de fonctions algébriques tout en augmentant le degré des places.En appliquant cette stratégie aux corps de fonctions elliptiques, on montre d'une part que le rang de tenseur de la multiplication dans $F_{q^n}$ est quasi-linéaire en $n$, et d'autre part que la construction des algorithmes de multiplications bilinéaires issus de cette stratégie est réalisable en temps polynomial. On montre également comment construire explicitement ces algorithmes sur $F_{q^n}$, en les illustrant par un exemple. Enfin, on établit la première construction asymétrique de l'algorithme de type Chudnovsky. / In this thesis, we focus on the bilinear complexity of multiplication in any degree $n$ extension of the finite field $ F_{q}$, which is closely related to the tensor rank of multiplication in $ F_{q^n} $. The evaluation-interpolation type algorithm introduced by D.V and G.V Chudnovsky in 1987, is the basis of all algorithmic technique providing for now, the lower asymptotic and uniform bounds for the bilinear complexity.So far, the strategy to obtain these lower bounds was to fix the degree of places while increasing the genus of algebraic function fields. However, the study of the effective construction associated with this kind of strategy was until now neglected because of an obstacle related to the construction of a degree $n$ point, identified par Shparlinski, Tsfasman and Vladut in 1992. We present a new strategy which consists in fixing the genus of algebraic function fields while increasing the degree of places. Applying this strategy to the elliptic function fields, we show on the one hand that the tensor rank of multiplication in $ F_{q^n} $ is quasi-linear in $ n $, and on the other hand we prove that the construction of bilinear multiplication algorithms with this strategy is feasible in polynomial time. We also show how to construct explicitly these algorithms over $ F_{q^n} $ for large $n$ by illustrating the construction with an example. Finally, we establish the first asymmetric construction of the Chudnovsky type algorithm.

Starting with the work of Volker Strassen, algorithms for matrix multiplication have been developed which are time complexity-wise more efficient than the standard algorithm from the definition of multiplication. The general method of the developments has been viewing the bilinear mapping that matrix multiplication is as a three-dimensional tensor, where there is an exact correspondence between time complexity of the multiplication algorithm and tensor rank. The latter can be seen as a generalisation of matrix rank, being the minimum number of terms a tensor can be decomposed as. However, in contrast to matrix rank there is no general method of computing tensor ranks, with many values being unknown for important three-dimensional tensors. To further improve the theoretical bounds of the time complexity of matrix multiplication, support rank of tensors has been introduced, which is the lowest rank of tensors with the same support in some basis. The goal of this master's thesis has been to go through the history of faster matrix multiplication, as well as specifically examining the properties of support rank for general tensors. In regards to the latter, a complete classification of rank structures of support classes is made for the smallest non-degenerate tensor product space in three dimensions. From this, the size of a support can be seen affecting the pool of possible ranks within a support class. At the same time, there is in general no symmetry with regards to support size occurring in the rank structures of the support classes, despite there existing a symmetry and bijection between mirrored supports. Discussions about how to classify support rank structures for larger tensor product spaces are also included. / Från och med forskning gjord av Volker Strassen har flera algoritmer för matrismultiplikation utvecklats som är effektivare visavi tidskomplexitet än standardalgoritmen som utgår från defintionen av multiplikation. Generellt sett har metoden varit att se den bilinjära avbildningen som matrismultiplikation är som en tredimensionell tensor. Där används att det finns en exakt korrespondens mellan multiplikationsalgoritmens tidskomplexitet och tensorrang. Det sistnämnda är ett slags generalisering av matrisrang, och är minsta antalet termer en tensor kan skrivas som. Till skillnad frpn matrisrang finns ingen allmän metod för att beräkna tensorrang, och många värden är okända även för välstuderade och viktiga tensorer. För att hitta fler övre begränsningar på matrismultiplikations tidskomplexitet har stödrang av tensorer införts, som är den lägsta rangen bland tensor med samma stöd i en viss bas. Målet med detta examensarbete har varit att göra en genomgång av historien om snabbare matrismultiplikation, samt att specifikt undersöka egenskaper av stödrang för allmänna tredimensionella tensorer. För det sistnämnda görs en fullständig klassificering av rangstrukturer bland stödklasser för den minsta icke-degenererade tensorprodukten av tre vektorrum. Slutsatser är bl.a. att storleken av ett stöd kan ses påverka antalet möjliga ranger inom en stödklass. Samtidigt finns i allmänhet ingen symmetri med avseende på stödstorlek i stödklassernas rangstrukturer. Detta trots att det finns en symmetri och bijektion mellan speglade stöd. I arbetet ingår även en diskussion om hur stödrangstrukturer skulle kunna klassificeras för större tensorprodukter.

On s'intéresse dans cette thèse à la détermination du rang de tenseur de la multiplication dans $mathbb{F}_{q^n}$, l'extension de degré $n$ du corps fini $mathbb{F}_q$ ; ce rang de tenseur correspond en particulier à la complexité bilinéaire de la multiplication dans $mathbb{F}_{q^n}$ sur $mathbb{F}_q$. Dans cette optique, on présente les différentes évolutions de l'algorithme de type évaluation-interpolation introduit en 1987 par D.V. et G.V. Chudnovsky et qui a permis d'établir que le rang de tenseur de la multiplication dans $mathbb{F}_{q^n}$ était linéaire en~$n$. Cet algorithme en fournit désormais les meilleures bornes connues dans le cas d'extensions de degré grand relativement au cardinal du corps de base — le cas des petites extensions étant bien connu. Afin d'obtenir des bornes uniformes en le degré de l'extension, il est nécessaire, pour chaque $n$, de déterminer un corps de fonctions algébriques qui convienne pour appliquer l'algorithme pour $mathbb{F}_{q^n}$, c'est-à-dire qui ait suffisamment de places de petit degré relativement à son genre $g$ et pour lequel on puisse établir l'existence de diviseurs ayant certaines propriétés, notamment des diviseurs non-spéciaux de degré ${g-1}$ ou de dimension nulle et de degré aussi près de ${g-1}$ que possible ; c'est pourquoi les tours de corps de fonctions sont d'un intérêt considérable. En particulier, on s'intéresse ici à l'étude des tours de Garcia-Stichtenoth d'extensions d'Artin-Schreier et de Kummer qui atteignent la borne de Drinfeld-Vlu{a}duc{t}. / In this thesis, we focus on the determination of the tensor rank of multiplication in $mathbb{F}_{q^n}$, the degree $n$ extension of the finite field $mathbb{F}_q$, which corresponds to the bilinear complexity of multiplication in $mathbb{F}_{q^n}$ over $mathbb{F}_q$. To this end, we describe the various successive improvements to the evaluation-interpolation algorithm introduced in 1987 by D.V. and G.V. Chudnovsky which shows the linearity of the tensor rank of multiplication in $mathbb{F}_{q^n}$ with respect to $n$. This algorithm gives the best known bounds for large degree extensions relative to the cardinality of the base field (the case when the degree of the extension is small is well known). In order to obtain uniform bounds, we need to determine, for each $n$, a suitable algebraic function field for the algorithm on $mathbb{F}_{q^n}$, namely a function field with sufficiently many places of small degree relative to its genus $g$ and for which we can prove the existence of divisors with some good properties such as non-special divisors of degree ${g-1}$ or zero-dimensional divisors with degree as close to ${g-1}$ as possiblestring; these conditions lead us to consider towers of algebraic function fields. In particular, we are interested in the study of Garcia-Stichtenoth towers of Artin-Schreier and Kummer extensions which attain the Drinfeld-Vlu{a}duc{t} bound.

<p>The question about how to determine the rank of a tensor has been widely studied in the literature. However the analytical methods to compute the decomposition of tensors have not been so much developed even for low-rank tensors.</p><p>In this report we present analytical methods for finding real and complex PARAFAC decompositions of 2 x 2 x 2 tensors before computing the actual rank of the tensor.</p><p>These methods are also implemented in MATLAB.</p><p>We also consider the question of how best lower-rank approximation gives rise to problems of degeneracy, and give some analytical explanations for these issues.</p>

The question about how to determine the rank of a tensor has been widely studied in the literature. However the analytical methods to compute the decomposition of tensors have not been so much developed even for low-rank tensors. In this report we present analytical methods for finding real and complex PARAFAC decompositions of 2 x 2 x 2 tensors before computing the actual rank of the tensor. These methods are also implemented in MATLAB. We also consider the question of how best lower-rank approximation gives rise to problems of degeneracy, and give some analytical explanations for these issues.

This master's thesis addresses numerical methods of computing the typical ranks of tensors over the real numbers and explores some properties of tensors over finite fields. We present three numerical methods to compute typical tensor rank. Two of these have already been published and can be used to calculate the lowest typical ranks of tensors and an approximate percentage of how many tensors have the lowest typical ranks (for some tensor formats), respectively. The third method was developed by the authors with the intent to be able to discern if there is more than one typical rank. Some results from the method are presented but are inconclusive. In the area of tensors over nite filds some new results are shown, namely that there are eight GLq(2) GLq(2) GLq(2)-orbits of 2 2 2 tensors over any finite field and that some tensors over Fq have lower rank when considered as tensors over Fq2 . Furthermore, it is shown that some symmetric tensors over F2 do not have a symmetric rank and that there are tensors over some other finite fields which have a larger symmetric rank than rank.

This dissertation reviews the theory of fast matrix multiplication from a multilinear-algebraic point of view, as well as recent fast matrix multiplication algorithms based on discrete Fourier transforms over nite groups. To this end, the algebraic approach is described in terms of group algebras over groups satisfying the triple product Property, and the construction of such groups via uniquely solvable puzzles. The higher order singular value decomposition is an important decomposition of tensors that retains some of the properties of the singular value decomposition of matrices. However, we have proven a novel negative result which demonstrates that the higher order singular value decomposition yields a matrix multiplication algorithm that is no better than the standard algorithm. / Mathematical Sciences / M. Sc. (Applied Mathematics)

This dissertation reviews the theory of fast matrix multiplication from a multilinear-algebraic point of view, as well as recent fast matrix multiplication algorithms based on discrete Fourier transforms over nite groups. To this end, the algebraic approach is described in terms of group algebras over groups satisfying the triple product Property, and the construction of such groups via uniquely solvable puzzles. The higher order singular value decomposition is an important decomposition of tensors that retains some of the properties of the singular value decomposition of matrices. However, we have proven a novel negative result which demonstrates that the higher order singular value decomposition yields a matrix multiplication algorithm that is no better than the standard algorithm. / Mathematical Sciences / M. Sc. (Applied Mathematics)