Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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The maximum likelihood degree of linear spaces of symmetric matrices
Full text linkWe 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 of symmetric matrices. We obtain new formulae for the ML degree, one via line geometry, and another using Segre classes from intersection theory. We settle the case of codimension one models, and characterize the degenerate case when the ML degree is zero
Perturbation and stability bounds for ergodic general state Markov chains with respect to various norms
Full text linkThis paper provides new perturbation bounds for general state Markov chains with respect to various norms. The transition and stationary char- acteristics estimates are given in terms of the generalized norm ergodic- ity coefficient, the norm of a residual kernel, or the parameters given in some drift condition. In fact, those results improve and generalize, for general state-space with respect to various norms and for a more large amplitude of perturbation of the transition kernel, the bounds obtained in [1, 32, 40]. Furthermore, we improve some other inequalities established in [22, 39, 41]. Theoretical comparison and on the basis of some examples show the quality of the results obtained in this paper
Pencils of quadrics: old and new
Full text linkTwo-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 reciprocal curve and the maximum likelihood degrees, and we study strata of pencils in the Grassmannian
The conditions for blow-up and global existence of solution for a degenerate and singular parabolic equation with a non-local source
Full text linkIn this paper, we consider the degenerate and singular porous medium equation with a non-local source The conditions on the local and global existence of solutions are investigated. In the case of blow-up, the blow-up set is shown. Moreover, the uniform blow-up profile of the blow-up solution is given
Equilibration in a two-species-two-chemicals chemotaxis-competition system
Full text linkThis paper deals with the two-species--two-chemicals chemotaxis-competition system with signal-dependent sensitivity,
where Ω is a bounded domain in Rn (n ≥ 2) with smooth boundary, χ1,χ2 and μ1,μ2 are constants satisfying some conditions. About this system Tu–Mu–Zheng–Lin (Discrete Contin. Dyn. Syst.;2018;38;3617– 3636) showed global existence and stabilization of solutions under some smallness conditions for χ1 and χ2. Here energy arguments for seeing stabilization in the previous work were based on ideas in Bai–Winkler (Indiana Univ. Math. J.;2016;65;553–583); however, these ideas were recently improved by the first author (Discrete Contin. Dyn. Syst. Ser. S;2020;13;269–278), which implies that the result about stabilization in the previous work seems not to be the best. This paper gives an improve- ment of conditions for stabilization in the previous work. The feature of the proof is to use the Sylvester criterion in deriving energy estimates
Nodes on quintic spectrahedra
Full text linkWe 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
Maximum likelihood estimation for nets of conics
Full text linkWe 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 Gaussian distributions on R3. In particular, we study the reciprocal surfaces of nets of conics which are rational surfaces in P5. We show that the reciprocal surfaces are projections from the Veronese surface and determine their intersection with the polar nets. This geometry explains the maximum likelihood degrees of these linear models. We compute the reciprocal maximum likelihood degrees. This work is based on Wall\u27s classification of nets of conics from 1977
The degree of the central curve in semidefinite, linear, and quadratic programming
Full text linkThe 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 complexity of these interior point algorithms, and for linear programs it was computed by De Loera, Sturmfels, and Vinzant in 2012. We show that the degree of the central curve for generic semidefinite programs is equal to the maximum likelihood degree of linear concentration models. New results from the intersection theory of the space of complete quadrics imply that this is a polynomial in the size of semidefinite matrices with degree equal to the number of constraints. Besides its degree we explore the arithmetic genus of the same curve. We also compute the degree of the central curve for generic linear programs with different techniques which extend to bounding the same degree for generic quadratic programs
Likelihood geometry of correlation models
Full text linkCorrelation 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 encode additional symmetries. We also consider the problem of minimizing two closely related functions of the covariance matrix: the Stein\u27s loss and the symmetrized Stein\u27s loss. Unlike the Gaussian log-likelihood these two functions are convex and hence admit a unique positive definite optimum. Some of our results hold for general affine covariance models
Preface
Full text linkThe 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 linear Gaussian covariance or concentration models, while in enumerative algebraic geometry they classically represent spaces of smooth quadrics satisfying certain tangency conditions. In semidefinite programming, linear spaces of symmetric matrices define the spectrahedra on which optimization problems are considered, and in nonlinear algebra they encode partially symmetric tensors.
It is often the case that one of the above-mentioned fields inspires or pro- vides tools for the advancement of the others. In the articles that follow, the reader will find several examples where this has happened through the common link of linear spaces of symmetric matrices.
This volume is the culmination of a collaboration project with the same name, which began at MPI Leipzig in June 2020. Over the course of several months, about 40 researchers gathered on-line to work on the ideas and projects that eventually became the articles of this special issue.
We are grateful to Bernd Sturmfels for initiating the project and for being its driving force, in particular for presenting the list of open problems that served as a starting point for the working groups that formed. Many of his conjectures became theorems in this volume.
We thank Biagio Ricceri and the editorial team of Le Matematiche for co- ordinating the publication of this volume. Finally, thanks to all participants for their contributions to the talks, discussions, and articles around the project