Algorithmes de diagonalisation conjointe par similitude pour la décomposition canonique polyadique de tenseurs : applications en séparation de sources

Abstract : This thesis introduces new joint eigenvalue decomposition algorithms. These algorithms allowamongst others to solve the canonical polyadic decomposition problem. This decomposition iswidely used for blind source separation. Using the joint eigenvalue decomposition to solve thecanonical polyadic decomposition problem allows to avoid some problems whose the others canonicalpolyadic decomposition algorithms generally suffer, such as the convergence rate, theoverfactoring sensibility and the correlated factors sensibility. The joint eigenvalue decompositionalgorithms dealing with complex data give either good results when the noise power is low, orthey are robust to the noise power but have a high numerical cost. Therefore, we first proposealgorithms equally dealing with real and complex. Moreover, in some applications, factor matricesof the canonical polyadic decomposition contain only nonnegative values. Taking this constraintinto account makes the algorithms more robust to the overfactoring and to the correlated factors.Therefore, we also offer joint eigenvalue decomposition algorithms taking advantage of thisnonnegativity constraint. Suggested numerical simulations show that the first developed algorithmsimprove the estimation accuracy and reduce the numerical cost in the case of complexdata. Our numerical simulations also highlight the fact that our nonnegative joint eigenvaluedecomposition algorithms improve the factor matrices estimation when their columns have ahigh correlation degree. Eventually, we successfully applied our algorithms to two blind sourceseparation problems : one concerning numerical telecommunications and the other concerningfluorescence spectroscopy.
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Rémi André. Algorithmes de diagonalisation conjointe par similitude pour la décomposition canonique polyadique de tenseurs : applications en séparation de sources. Traitement du signal et de l'image [eess.SP]. Université de Toulon, 2018. Français. ⟨NNT : 2018TOUL0011⟩. ⟨tel-02278358⟩

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