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Exponential varieties. (English) Zbl 1345.14048
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.
Exponential 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 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 \(f\).
A 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 \(C\), called the hyperbolicity cone of \(f\). It is known from optimization theory that the gradient function \(F(\theta)=-\log f(\theta)\) provides a bijection between the hyperbolicity cone \(C\) and the interior of its dual cone \(K=C^{\vee}\) [J. Renegar, MPS/SIAM Series on Optimization 3. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. Philadelphia, PA: MPS, Mathematical Programming Society, vi, 116 p. (2001; Zbl 0986.90075)].
The paper is organized as follows. In Section 2, exponential families are introduced. These are particular parametric statistical models with a well-known duality between the space \(C\) of canonical parameters and \(K\) of sufficient statistics [L. D. Brown, Fundamentals of statistical exponential families with applications in statistical decision theory. Hayward, CA: Institute of Mathematical Statistics (1986; Zbl 0685.62002)]. Section 3 is devoted to describing the hyperbolic exponential family associated to a hyperbolic polynomial \(f\). These are exponential families whose space of canonical parameters coincides with the hyperbolicity cone \(C\) of \(f\).
In Section 4 exponential varieties are defined. A key result capturing the strong convexity and positivity properties of exponential varieties is found in Theorem 4.4, which we now describe. Suppose \(\mathcal{L}\subset\mathbb{R}^d\) is a linear subspace and \(f\in \mathbb{R}[x_1,\ldots,x_d]\) is a hyperbolic polynomial with hyperbolicity cone \(C\) and gradient map \(F=-\nabla\log(f)\). Let \(\mathcal{L}^F\) be the associated exponential variety. The positive part of \(\mathcal{L}^F\) is the semi-algebraic set \(\mathcal{L}^F_{\succ 0}=F(C\cap \mathcal{L})\subset K=C^\vee\).
Dual to the inclusion of the linear subspace \(\mathcal{L}\subset \mathbb{R}^d\) is the projection \(\pi_{\mathcal{L}}:\mathbb{R}^d\rightarrow (\mathbb{R}^d/\mathcal{L}^{\perp}) \mathbb{R}^c\) with kernel \(\mathcal{L}^\perp\). Set \(C\cap\mathcal{L}=C_\mathcal{L}\) and \(K_\mathcal{L}=L(K)\). Theorem 4.4 says that in the sequence of maps \[ C_\mathcal{L}\subset C\rightarrow{F} K\rightarrow{L} K_{\mathcal{L}}, \] \(C_\mathcal{L}\) is mapped bijectively to \(\mathcal{L}^F_{\succ 0}\subset K\) by \(F\) and \(\mathcal{L}^F_{\succ 0}\) is mapped bijectively to \(K_{\mathcal{L}}\) by the projection \(\pi_{\mathcal{L}}\). The projection \(\pi_{\mathcal{L}}:\mathcal{L}^F_{\succ 0}\rightarrow K_{\mathcal{L}}\) is the analog of the moment map for toric varieties.
Section 5 is devoted to studying the degree of the projection \(\pi_\mathcal{L}\). In Section 6, exponential varieties defined by symmetric polynomials are studied, and in Section 7 the inverses of Hankel matrices satisfying linear constraints are studied.

14M99 Special varieties
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
62E99 Statistical distribution theory
65C60 Computational problems in statistics (MSC2010)
Bertini; gRc; Macaulay2
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