Computing convex hulls of trajectories

Ciripoi, Daniel; Kaihnsa, Nidhi; Löhne, Andreas; Sturmfels, Bernd

doi:10.33044/revuma.v60n2a22

Cited by 7 publications

(9 citation statements)

References 14 publications

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“…In the discrete setting, it was done for linear models [1]. In the Gaussian setting, combinatorial types of spectrahedra can be described using patches; see [7,15].…”

Section: Undirected Graphical Modelsmentioning

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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“…In the discrete setting, it was done for linear models [1]. In the Gaussian setting, combinatorial types of spectrahedra can be described using patches; see [7,15].…”

Section: Undirected Graphical Modelsmentioning

confidence: 99%

Logarithmic Voronoi Cells for Gaussian Models

Yulia¹,

Hoşten²

2022

Preprint

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“…Comparison with previous work. In [4], the authors suggest an algorithm in Section 5 (Algorithm 5.4) to detect the number of patches for the convex hull of a sufficiently nice curve which they call simplicial. First of all, their definition of a patch is slightly different from ours.…”

Section: 1mentioning

confidence: 99%

“…Their proposed algorithm [4,Algorithm 5.4] can be viewed as a way to sample from N (conv(C)) and the patches without knowing the algebraic boundary of K = conv(C). Starting with a finite sample of points on C, they first compute the convex hull of these finitely many points and record the incidences of this polytope (step 1).…”

Section: 1mentioning

confidence: 99%

“…Spectrahedra (and more generally hyperbolicity cones) are central objects at the intersection of convex algebraic geometry and optimization [2] [11]. Convex hulls of real algebraic sets appear, for instance, in the context of attainable regions of dynamical systems [4] and in the study of quantum systems [8] and have been studied in classical projective geometry [9] [10].…”

Section: Introductionmentioning

confidence: 99%

“…A computational approach to identify families of faces for convex hulls of curves was presented in [4] where the authors introduce the concept of a patch and give a numerical algorithm to compute the boundary structure. We propose a definition of a patch based on semi-algebraic geometry rather than using analytical notions and derive the main analytical features from our geometric definition.…”

Section: Introductionmentioning

confidence: 99%

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Families of faces and the normal cycle of a convex semi-algebraic set

Plaumann¹,

Sinn²,

Wesner³

2021

Preprint

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We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a patch -a term recently coined by Ciripoi, Kaihnsa, Löhne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences.

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Fiber Convex Bodies

Mathis

Meroni

2022

Discrete Comput Geom

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In this paper we study the fiber bodies, that is the extension of the notion of fiber polytopes for more general convex bodies. After giving an overview of the properties of the fiber bodies, we focus on three particular classes of convex bodies. First we describe the strict convexity of the fiber bodies of the so called puffed polytopes. Then we provide an explicit equation for the support function of the fiber bodies of some smooth convex bodies. Finally we give a formula that allows to compute the fiber bodies of a zonoid with a particular focus on certain zonoids called discotopes. Throughout the paper we illustrate our results with detailed examples.

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Computing convex hulls of trajectories

Cited by 7 publications

References 14 publications

Logarithmic Voronoi Cells for Gaussian Models

Logarithmic Voronoi Cells for Gaussian Models

Families of faces and the normal cycle of a convex semi-algebraic set

Fiber Convex Bodies

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