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Casarotti, Alex, (2021)  - Difettivitą e Identificabilitą: un punto di vista geometrico sull'analisi tensoriale.  - , Tesi di dottorato - (, , Universitą degli studi di Modena e Reggio Emilia ) - pagg. -

Abstract: Given a projective variety $X \subset \mathbb{P}^N$ we can define its $h-$secant variety $\mathbb{S}ec_h(X) \subset \mathbb{P}^N$, i.e. the Zariski closure of all points $p \in \mathbb{P}^N$ lying on a $\mathbb{P}^{h-1}$ which is $h-$secant to $X$. The variety $X$ is said to be $h-$identifiable if the general point $p \in \mathbb{S}ec_h(X)$ can be expressed uniquely as a linear combination $p=\lambda_1p_1+\dots+\lambda_hp_h$ with $p_1,\dots,p_h$ points of $X$. Thanks to Terracini's lemma it is possible to rephrase the problem of secant dimensions and identifiability in the birational setting. This turns out in the study of the dimension and the singularities of linear systems of the form $|\mathcal{O}_X(1)\otimes \mathcal{I}_{p_1^2,\dots,p_h^2}|$, i.e. hyperplane divisor of $X$ singular at $h$ general points. In the area of tensor analysis these notions are related to the properties of tensor decomposition. For applications ranging from biology to Blind Signal Separation, data compression algorithms and analysis of mixture models, uniqueness of decompositions allows to solve the problem once a solution is determined. The thesis studies the relation between defectiveness and identifiability. It is shown how to link the geometry of the tangential contact locus to the secant defect, proving that under mild numerical conditions the non $h-$secant defectiveness imply the $(h-1)-$identifiability, where $h$ is less than the generic rank. With our techinques it is possible to give new bounds for the identifiability in the case of many important tensor varieties such as Veronese, Segre and Grassmannians. In the case of generic identifiability it is studied the nested singularities of tangential linear system. With this, together with the classical Noether-Fano inequalities, it is proved a new statement on generic identifiability of many partially symmetric tensors. Next it is studied the defectiveness for Flag varieties, i.e. special tensor varieties parametrizing chains of vector spaces $0 \subset V_1 \subset \dots \subset V_k \subset \mathbb{P}^N$. We improve the osculating projection techinque from Araujo, Massarenti and Rischter, giving completely new bounds on secant defectiveness and identifiability. The new notion of $(h,s)-$tangential weak defectiveness is introduced and studied for the case of Segre-Veronese varieties. The study of Secant varieties of Veronese embedding allowed also to check Comon's conjecture under improved numerical bounds.