Conference Agenda

Tensors are ubiquitous in mathematics and science. Tensor decompositions and approximations are an important tool in artificial intelligence, chemometrics, complexity theory, signal processing, statistics, and quantum information theory. Often, due to the nature of the problem under investigation, it might be natural to consider tensors equipped with additional structures or might be useful to consider tensor decompositions which respect particular structures. Among many interesting constructions, we might think of: symmetric, partially-symmetric and skew-symmetric tensors; tensor networks; Hadamard products of tensors or non-negative ranks. This minisymposium focuses on how exploiting these additional structures from algebraic and geometric perspectives recently gave new tools to study these special classes of tensors and decompositions. This is a sister minisymposium to "The algebra and geometry of tensors 1: general tensors" organized by Y. Qi and N. Vannieuwenhoven.

For a point p and a variety X in a projective space, the X -rank of p is the minimal n such that p lies on an n-secant (n-1)-space to X. I consider the variety V(p,X) of such n-secants, will recall the cases when X is a rational or elliptic curve and explain some surface cases. A very nice result of Iliev and Manivel for tensors C^3 x C^3 have applications on these issues.

The maximal minors of the generic k x n Hankel matrix (also known as the catalacticant matrix) defines a rational map from the projective n-space to the Grassmann variety of (k-1)-planes in the projective n-space. The image of the projective n-space under the rational map is called the Hankel variety of (k-1)-planes, which is birationally equivalent to the projective n-space if n is sufficiently large compared with k. This talk concerns higher secant varieties of the Hankel variety of (k-1)-planes. The main focus of the talk is set on the defectivity of the Hankel variety of lines.

Tensor decomposition problems appear in many areas such as Signal Processing, Quantum Information Theory, Algebraic Statistics, Biology, Complexity Analysis, etc as a way to recover hidden structures from data. The decomposition is a representation of the tensor as a weighted sum of a minimal number of indecomposable terms. This problem can be seen as a sparse recovery problem from sequences of moments. We will develop this analogy and present an algebraic approach to address the decomposition problem, via duality and Hankel operators. We will analyze the varieties of moments associated to low rank decompositions, investigate their defining equations and some of their properties that can be exploited in the decomposition problem. Links with the Hilbert scheme of points will be presented. Examples exploiting these properties will illustrate the approach.

Let X be a Segre-Veronese product of projective spaces and denote with X* its dual variety. In this talk, we outline the main properties of the ``Euclidean Distance polynomial'' (ED polynomial) of X*, as a remarkable example of a more general theory on ED polynomials developed in a recent work with Ottaviani. Given a tensor T, the roots of the ED polynomial of X* at T correspond to the singular values of T. Moreover, we describe the variety of tensors that fail to have the expected number of singular vector tuples, counted with multiplicity. This variety is, in general, a non-reduced hypersurface and its equation is, up to scalars, the leading coefficient of the ED polynomial of X*.