Exponential varieties
type: Item
instance of: scholarly article
title: Exponential varieties
zbMATH Open document ID: 1345.14048
author: Mateusz Michalek
author: Bernd Sturmfels
author: Caroline Uhler
author: Piotr Zwiernik
publication date: 2016-03-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1412.6185
Mathematics Subject Classification ID: 14M99
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 62E99
Mathematics Subject Classification ID: 65C60
zbMATH DE Number: 6559631
zbMATH Keywords: algebraic statistics
zbMATH Keywords: semidefinite programming
zbMATH Keywords: convex optimization
zbMATH Keywords: exponential families
zbMATH Keywords: hyperbolic polynomials
zbMATH Keywords: ML degree
describes a project that uses: Bertini
describes a project that uses: Macaulay2
describes a project that uses: gRc
MaRDI profile type: Publication
OpenAlex ID: W3037491991
arXiv ID: 1412.6185
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DOI: 10.1112/PLMS/PDV066
published in: Proceedings of the London Mathematical Society. Third Series
review text: The authors define and develop the theory of exponential varieties, which, as they state in the abstract, are ``real algebraic varieties exhibiting strong convexity and positivity properties familiar from toric varieties and their moment maps.'' In fact, one can `almost' realize projective toric varieties as exponential varieties (Example~2.4 and Example~4.3, see also Remark~3). Many varieties (particularly of importance in algebraic statistics) fall into this family of exponential varieties. This includes what the authors call the paradigm of an exponential variety, namely the variety of inverses of symmetric matrices satisfying linear constraints, which is discussed in detail for Hankel matrices in Section~7.NEWLINENEWLINEExponential varieties are defined in Section~4 as the closure of the image of a linear subspace under the gradient map of a hyperbolic polynomial. More precisely, given a linear space \(\mathcal{L}\subset \mathbb{C}\mathbb{P}^{d-1}\) and a homogeneous polynomial \(f\in \mathbb{R}[\theta_1,\ldots,\theta_d]\) with gradient map \(F=-\nabla\log(f):\mathbb{C}\mathbb{P}^{d-1}\dashrightarrow \mathbb{C}\mathbb{P}^{d-1}\), one can always consider the variety \(\mathcal{L}^{F}\subset \mathbb{C}\mathbb{P}^{d-1}\) which is the closure of the image of \(\mathcal{L}\) under \(F\). If the polynomial is in addition assumed to be \textit{hyperbolic}, then \(\mathcal{L}^{\nabla f}\) is an exponential variety. The positivity properties of an exponential variety, which are not at all evident from this definition, follow from the hyperbolicity of \(f\).NEWLINENEWLINEA homogeneous polynomial \(f\in \mathbb{R}[\theta_1,\ldots,\theta_d]\) of degree \(p\) is hyperbolic if there exists a point \(\tau\in\mathbb{R}^d\) so that \(f(\tau)\neq 0\), satisfying that every line through \(\tau\) intersects the real hypersurface \(\{f=0\}\) in \(p\) points (counting multiplicity). The component of \(\mathbb{R}^d\setminus \{f=0\}\) that contains \(\tau\) is a convex cone \(C\), call…
reviewed by: Michael R. DiPasquale
description: scientific article; zbMATH DE number 6559631