Voronoi cells of varieties

Cifuentes, Diego; Ranestad, Kristian; Sturmfels, Bernd; Weinstein, Madeleine

doi:10.1016/j.jsc.2020.07.009

Cited by 9 publications

(9 citation statements)

References 16 publications

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“…1 +4x 2 2 +4x 2 3 = 4. We find that its ED discriminant is of degree 6 and it is defined by the polynomial 64y 6 1 +48y 4 1 y 2 2 +12y 2 1 y 4 2 +. .…”

Section: Curv=saturate(ideal(edpoly Df G) Ideal(e));mentioning

confidence: 99%

“…As δ goes to there are two distinct cases. One, if x 1 δ → x 1 and x 2 δ → x 2 , with x 1 = x 2 , then y 0 is at the intersection of two distinct normal lines with d(y 0 , x 1 ) = d(y 0 , x 2 ), hence being an element of the bisector hypersurface to X (see for instance [13][Section 2.3]) or in other words of the symmetry set of X or the union of all Voronoi boundaries, see for instance [6]. Let us denote the bisector hypersurface by B(X, X).…”

Section: Oc(x)mentioning

confidence: 99%

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The critical curvature degree of an algebraic variety

Horobet¹

2021

Preprint

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“…1 +4x 2 2 +4x 2 3 = 4. We find that its ED discriminant is of degree 6 and it is defined by the polynomial 64y 6 1 +48y 4 1 y 2 2 +12y 2 1 y 4 2 +. .…”

Section: Curv=saturate(ideal(edpoly Df G) Ideal(e));mentioning

confidence: 99%

Section: Oc(x)mentioning

confidence: 99%

The critical curvature degree of an algebraic variety

Horobet¹

2021

Preprint

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“…For any U ⊆ R m and p ∈ U , the Euclidean Voronoi cell at p is the set of all points in R m that are closer to p than any other point in U with respect to the Euclidean metric. Euclidean Voronoi cells of varieties were studied in [6] and are the topic in metric algebraic geometry [8,9,21]. In general, logarithmic Voronoi cells are not equal to Euclidean Voronoi cells.…”

Section: Basics Of Gaussian Models and Logarithmic Voronoi Cellsmentioning

confidence: 99%

“…If ∆ f (b, a) > 0 and a < 1/2, then we must compare the value that n takes on Σ c to the values that it takes on the two matrices Σ 1 , Σ 2 corresponding to the other two real roots of f (x), for the fixed a and b. Given the relationship between a and b as in (6) we find all three roots of f (x) in terms of b and c. They are…”

Section: Bivariate Correlation Modelmentioning

confidence: 99%

Logarithmic Voronoi Cells for Gaussian Models

Yulia¹,

Hoşten²

2022

Preprint

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We extend the theory of logarithmic Voronoi cells to Gaussian statistical models. In general, a logarithmic Voronoi cell at a point on a Gaussian model is a convex set contained in its log-normal spectrahedron. We show that for models of ML degree one and linear covariance models the two sets coincide. In particular, they are equal for both directed and undirected graphical models. We introduce decomposition theory of logarithmic Voronoi cells for the latter family. We also study covariance models, for which logarithmic Voronoi cells are, in general, strictly contained in log-normal spectrahedra. We give an explicit description of logarithmic Voronoi cells for the bivariate correlation model and show that they are semi-algebraic sets. Finally, we state a conjecture that logarithmic Voronoi cells for unrestricted correlation models are not semi-algebraic.

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“…When X is a finite set, Voronoi cells at all points in X tessellate R n into convex polyhedra. When X is a variety, the Voronoi cells of X are convex semialgebraic sets in the normal space of X, and their algebraic boundary was computed in [5]. One can study the properties of real algebraic varieties, such as their Voronoi cells, that depend on the distance metric.…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Voronoi polytopes for discrete linear models

Yulia¹

2021

Preprint

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We study logarithmic Voronoi cells for linear statistical models and partial linear models. The logarithmic Voronoi cells at points on such model are polytopes. To any d-dimensional linear model inside the probability simplex ∆ n−1 , we can associate an n×d matrix B. For interior points, we describe the vertices of these polytopes in terms of co-circuits of B. We also show that these polytopes are combinatorially isomorphic to the dual of a vector configuration with Gale diagram B. This means that logarithmic Voronoi cells at all interior points on a linear model have the same combinatorial type. We also describe logarithmic Voronoi cells at points on the boundary of the simplex. Finally, we study logarithmic Voronoi cells of partial linear models, where the points on the boundary of the model are especially of interest.

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Voronoi cells of varieties

Cited by 9 publications

References 16 publications

The critical curvature degree of an algebraic variety

The critical curvature degree of an algebraic variety

Logarithmic Voronoi Cells for Gaussian Models

Logarithmic Voronoi polytopes for discrete linear models

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