Presentations

Hitchcock decompositions of tensors (aka Canonical Polyadic Decompositions or PARAFAC Decomposition) encode mixtures of independence models on discrete random variables, namely, statistical models given by one hidden discrete variable connected to a set of observed ones. Hadamard products, i.e., entry-wise products, of Hitchcock decompositions encode products of mixtures of independence models on discrete random variables, namely, statistical models given by a complete bipartite graph with a layer of hidden discrete variables and a layer of observed ones (aka Restricted Boltzmann Machines). In this talk, after recalling some recent literature about these models, I will present a new generic identifiability result of such decompositions (under numerical assumptions on the number of states). Our approach also suggests an algorithm to compute a Hadamard-Hitchcock decomposition of a generic tensor.

- Hadamard-Hitchcock Decompositions: identifiability and computation, with N. Vannieuwenhoven, arXiv:2308.06597, 2023

We consider the inverse problem for the polynomial map which sends an m-tuple of quadratic forms in n variables to the sum of their d-th powers. This map captures the moment problem for mixtures of m centered n-variate Gaussians. In the first non-trivial case d=3, we show that for any n∈ℕ, this map is generically one-to-one (up to permutations of q1,…,qm and third roots of unity) in two ranges: m≤(n2)+1 for n≤16 and m≤(n+56)/(n+12)−(n+12)−1 for n>16, thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most 6. The first result is obtained by studying the explicit geometry of the tangential contact locus of the variety of sums of cubes of quadratic forms at concrete points, while the second result is accomplished using a link between secant non-defectivity with identifiability. The latter approach generalizes also to sums of d-th powers of k-forms for d≥3 and k≥2.

- Identifiability for mixtures of centered Gaussians and sums of powers of quadratics, with A. Taveira Blomenhofer, A. Casarotti, M. Michałek, arXiv:2204.09356, 2022

A hyperplane defined by {L1 = ... = Lr = 0} contained in a hypersurface {F = 0} in projective space determines an expression of F = L1G1 + ... + LrGr. The strength of the polynomial F is the smallest length of an expression F = G1H1 + ... + GrHr.  In a joint work with Bik, Ballico and Ventura, we prove that strength and slice rank of homogeneous polynomials of degree d≥5 over an algebraically closed field of characteristic zero coincide generically. To show this, we establish a conjecture of Catalisano, Geramita, Gimigliano, Harbourne, Migliore, Nagel and Shin concerning dimensions of secant varieties of the varieties of reducible homogeneous polynomials. These statements were already known in degrees 2≤d≤7 and d=9 from a previous collaboration with Bik.

In this presentation, I describe the problem, the idea of the proof and its connections also with the theory of secant varieties and with the Froberg's conjecture on the Hilbert series of general ideals. 

This is base on a joint work:

- On the strength of general polynomials, with Arthur Bik, Linear and Multilinear Algebra
- Strength and slice rank of polynomials are generically equal, with Arthur Bik, Edoardo Ballico and Emanuele Ventura, Israel J. Math.

Secant defectivity of projective varieties is classically approached via dimensions of linear systems with multiple base points in general position. The latter can be studied via degenerations. We exploit a technique that allows some of the base points to collapse together. We deduce a general result which we apply to prove a conjecture by Abo and Brambilla: for c≥3 and d≥3, the Segre-Veronese embedding of P^m × P^n in bidegree (c,d) is non-defective.

This is based on a joint work:
- Secant non-defectivity of Segre-Veronese varieties via collapsing points, with Francesco Galuppi, Advances Math.

Given a tensor, i.e., an multi-dimensional array of numbers, we define its tensor rank as the smallest length of an additive decomposition of the tensor as a linear combination of decomposable tensors, i.e., tensors that can be expressed as tensor products of vectors. This is a generalisation of the notion of rank of matrices, which are simply 2-dimensional tensors. If the tensor is symmetric, i.e., it can be identified with a homogeneous polynomial, we may consider its symmetric rank as the smallest length of an additive decomposition of the tensor as a linear combination of decomposable symmetric tensors. 

Comon's problem (2008) asks wether the tensor rank and the symmetric tensor rank of a symmetric tensor coincide. In the past years, there have been positive answers for small dimensional symmetric tensors, but Shitov gave a negative answer by constructing a tensor of size 800x800x800 having tensor rank equal to 903, but symmetric rank 904. 

We also have the notion of partially symmetric tensors and partially symmetric rank and we can ask all partially symmetric versions of Comon's problem.

In this presentation, I describe this problem and I will relate it to the problem of finding the simultaneous rank of the partial derivatives of a homogeneous polynomials. 

This is base on a joint work:
- Variants of Comon's problem via simultaneous ranks, with Fulvio Gesmundo and Emanuele Ventura, SIMAX 

In this presentation, I introduce the concept of Waring locus of a homogeneous polynomial, i.e., the set of linear forms that can be used to write a minimal Waring decomposition of the polynomial. I show how to compute these loci in specific cases by using algebraic tools as apolarity theory and I explain how to use them to get a geometric algorithm to computes a minimal Waring decomposition of a given polynomial. 

This is based on two joint works:
- Waring loci and Strassen's conjecture, with Enrico Carlini and Marvi Catalisano, Advances Math.
- On minimal decompositions of low rank symmetric tensors, with Bernard Mourrain, Linear Alg. Applications

In this presentation, I want to present a recent question raised by Cook II, Harbourne, Migliore and Nagel (Line arrangements and configurations of points with an unusual geometric property, Comp. Math. 154 (2018) 2150-2194) on planar polynomial interpolation: given a set of reduced points Z, how many curves of degree m+1 pass through Z and a general m-fat point? When the expected number, given by a direct count of parameters, is not attained (and the number of curves is higher), we say that Z admits unexpected curves. The property of admitting unexpected curves is related to the exponent of the line arrangement dual to Z, i.e., (if we work in characteristic zero) the splitting type of the syzygy bundle of the Jacobian ideal of the polynomial defining the line arrangement. After the introduction of the problem, I present a classification of super-solvable line arrangements whose dual set of points admits unexpected curves.

This is based on a joint work:
- Unexpected curves arising from special line arrangements, with Michela Di Marca and Grzegorsz Malara, J. Alg. Combinatorics

In this presentation, I want to talk about interpolation problems over multi-projective spaces. A very classical question in algebraic geometry asks what is the dimension of the linear system of curves of degree d having multiple base points with support in general position. In more algebraic language, what is the Hilbert function in degree d of a scheme of fat points having support in general position. A conjectural answer is given by the famous SHGH Conjecture (Segre-Harbourne-Gimigliano-Hirschowitz Conjecture) which has been proved to be true for small number of points or small multiplicities. Here, we consider the bi-graded case by considering the Hilbert function of fat points on P^1 x P^1. The case of double points was completely solved by a previous work of Catalisano, Geramita and Gimigliano. Here, I explain the methods that lead us to solve the case of triple points and give partial results for any multiplicity. 

This is based on a joint work:
- Hilbert function of general fat points in P^1 x P^1, with Enrico Carlini and Maria Virginia Catalisano, Michigan J. of Math.