The Distance Function from the Variety of partially symmetric rank-one Tensors

The topic of this doctoral thesis is at the intersection between Real Algebraic Geometry, Optimization Theory and Multilinear Algebra. In particular, a relevant part of this thesis is dedicated to studying metric invariants of real algebraic varieties, with a particular interest in varieties in tensor spaces. In many applications, tensors arise as a useful way to store and organize experimental data. For example, it is widely known that tensor techniques are extremely useful in Algebraic Statistics. A strong relationship between classical algebraic geometry and multilinear algebra is established by the notion of tensor rank. Geometrically speaking, the problem of computing the rank of a tensor translates to a membership problem to a certain secant variety of a Segre product of projective spaces. In the last fifteen years, a new line of research in tensor theory has been undertaken and is commonly known as Spectral Theory of Tensors. One of the foundational motivations of this theory comes from the need, e.g., in some constrained optimization problems, to approximate a given tensor to its closest tensor of fixed lower rank, with respect to the Frobenius norm, also known as Bombieri norm. This is the so-called best rank- k approximation problem for real tensors. In this context, an important role is played by the singular vector tuples and the singular values of a tensor, which generalize the notions of eigenvector and eigenvalue of a matrix. Their symmetric counterpart is represented by the E-eigenvalues and the E-eigenvectors of a symmetric tensor. Of particular interest is the E-characteristic polynomial of a symmetric tensor, which has among its roots the E-eigenvalues of a symmetric tensor. For symmetric matrices, it coincides with the classical characteristic polynomial. We interpret the E-characteristic polynomial as an algebraic relation satisfied by the Frobenius distance between an assigned symmetric tensor and the dual affine cone of a Veronese variety. We show that the E-characteristic polynomial is monic only in the symmetric matrix case. We provide a rational formula for the product of the singular values of a partially symmetric tensor of hypercubic format. The formula generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues. This is the only case where no denominator occurs in the formula. Computing the distance from a variety of low-rank tensors is an important instance of a more general problem: computing the distance from a real algebraic variety X in a Euclidean space (V,q). We introduce a polynomial, called Euclidean Distance polynomial of X, which, for any data point u in V, has among its roots the distance ε from u to X. The ε^2-degree of the ED polynomial is the known Euclidean Distance degree of X. When X is transversal to the isotropic quadric Q={q=0}, we show that the ED polynomial of X is monic and we describe its lowest term completely.