Le Matematiche  

O. Marigliano, M. Michalek, K. Ranestad, T. Seynnaeve

The aim of this volume is to advance the understanding of linear spaces of symmetric matrices. These seemingly simple objects play many different roles across several fields of mathematics.For instance, in algebraic statistics these spaces appear as linea... see more

C. Fevola, Y. Mandelshtam, B. Sturmfels

Two-dimensional linear spaces of symmetric matrices are classified by Segre symbols. After reviewing known facts from linear algebra and projective geometry, we address new questions motivated by algebraic statistics and optimization. We compute the recip... see more

A. Bik, H. Eisenmann, B. Sturmfels

We study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian.

T. Brysiewicz, C. Fevola, B. Sturmfels

We examine quadratic surfaces in 3-space that are tangent to nine given figures. These figures can be points, lines, planes or quadrics. The numbers of tangent quadrics were determined by Hermann Schubert in 1879. We study the associated systems of polyno... see more

Y. Cid-Ruiz

We compute the equations and multidegrees of the biprojective variety that parametrizes pairs of symmetric matrices that are inverse to each other. As a consequence of our work, we provide an alternative proof for a result of Manivel, Micha\l{}ek, Mo... see more

T. Boege, J. I. Coons, C. Eur, A. Maraj, F. Roettger

We give an explicit formula for the reciprocal maximum likelihood degree of Brownian motion tree models. To achieve this, we connect them to certain toric (or log-linear) models, and express the Brownian motion tree model of an arbitrary tree as a toric f... see more

S. Dye, K. Kohn, F. Rydell, R. Sinn

We study the problem of maximum likelihood estimation for 3-dimensional linear spaces of 3 x 3 symmetric matrices from the point of view of algebraic statistics where we view these nets of conics as linear concentration or linear covariance models of Gaus... see more

T. Brysiewicz, K. Kozhasov, M. Kummer

We classify transversal quintic spectrahedra by the location of 20 nodes on the respective real determinantal surface of degree 5. We identify 65 classes of such surfaces and find an explicit representative in each of them.

A. Al Ahmadieh, M. Kummer, M. S. Sorea

To any homogeneous polynomial h we naturally associate a variety Oh which maps birationally onto the graph Gh of the gradient map ?h and which agrees with the space of complete quadrics when h is the determinant of the generic symmetric matrix. We gi... see more

C. Eur, T. Fife, J. A. Samper, T. Seynnaeve

We show that the reciprocal maximal likelihood degree (rmld) of a diagonal linear concentration model L ? Cn of dimension r is equal to (-2)r?M(1/2), where ?M is the characteristic polynomial of the matroid M associated to L. In particular, this... see more

Y. Jiang, K. Kohn, R. Winter

We study the maximum likelihood degree of linear concentration models in algebraic statistics. We relate the geometry of the reciprocal variety to that of semidefinite programming. We show that the Zariski closure in the Grassmanian of the set of linear s... see more

S. Hosten, I. Shankar, A. Torres

The Zariski closure of the central path which interior point algorithms track in convex optimization problems such as linear, quadratic, and semidefinite programs is an algebraic curve. The degree of this curve has been studied in relation to the complexi... see more

I. Davies, O. Marigliano

Coloured graphical models are Gaussian statistical models determined by an undirected coloured graph. These models can be described by linear spaces of symmetric matrices. We outline a relationship between the symmetries of the graph and the linear f... see more

L. Brustenga i Moncusí, E. Cazzador, R. Homs

C. Améndola, L. Gustafsson, K. Kohn, O. Marigliano, A. Seigal

We study multivariate Gaussian models that are described by linear conditions on the concentration matrix. We compute the maximum likelihood (ML) degrees of these models. That is, we count the critical points of the likelihood function over a linear space... see more

C. Améndola, P. Zwiernik

Correlation matrices are standardized covariance matrices. They form an affine space of symmetric matrices defined by setting the diagonal entries to one. We study the geometry of maximum likelihood estimation for this model and linear submodels that enco... see more