Related papers: Tropical toric maximum likelihood estimation
The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseux-valued ``lift'' of this point in the…
We establish connections between invariant theory and maximum likelihood estimation for discrete statistical models. We show that norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We…
We establish faithful tropicalisation for point configurations on algebraic tori. Building on ideas from enumerative geometry, we introduce tropical scaffolds and use them to construct a system of modular fan structures on the tropical…
Maximum likelihood estimation (MLE) is a fundamental problem in statistics. Characteristics of the MLE problem for discrete algebraic statistical models are reflected in the geometry of the $\textit{likelihood correspondence}$, a variety…
In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let $K$ be a complete non-Archimedean field, and let $X$ be a closed subscheme of a toric variety over $K$. We define the…
In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study…
Enumerative algebraic geometry deals with problems of counting geometric objects defined algebraically, An important class of enumerative problems is that of counting curves: given a class of curves in some projective variety defined by…
We propose some problems on the classification of toric manifolds from the viewpoint of topology and survey related results.
Given a model in algebraic statistics and some data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying…
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…
In analogy to chapter 9 of arXiv:0709.3705 we define an intersection product of tropical cycles on tropical linear spaces L^n_k, i.e. on tropical fans of the type max{0,x_1,...,x_n}^(n-k)*R^n. Afterwards we use this result to obtain an…
Sparse polynomial systems with vertical coefficient dependencies arise naturally when describing the critical points of optimization problems and, when augmented with linear forms, the steady states of chemical reaction networks. Moreover,…
We study representations of tropical linear spaces as intersections of tropical hyperplanes of circuits. For several classes of matroids, we describe minimal tropical bases. We also show that every realizable tropical linear space has a…
A polynomial complexity algorithm is designed which tests whether a point belongs to a given tropical linear variety.
We study the problem of maximizing information divergence from a new perspective using logarithmic Voronoi polytopes. We show that for linear models, the maximum is always achieved at the boundary of the probability simplex. For toric…
We present two effective tools for computing the positive tropicalization of algebraic varieties. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to…
We find a relation between mixed volumes of several polytopes and the convex hull of their union, deducing it from the following fact: the mixed volume of a collection of polytopes only depends on the product of their support functions…
We study tropical degree bounds, stable tropical intersections, and tropical B\'ezout-type estimates through the geometry of Newton polytopes, mixed subdivisions, and lattice indices. We establish an upper bound for the tropical degree of a…
We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals…
For a complex hypersurface of dimension $d \geq 1$ in a toric variety, we construct lifts of tropical $(p, q)$-cycles with $p+q=d$ in the associated tropical hypersurface. The tropical cycles we consider are described by Minkowski weights,…