×

zbMATH — the first resource for mathematics

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.

MSC:
14M99 Special varieties
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
62E99 Statistical distribution theory
65C60 Computational problems in statistics (MSC2010)
Software:
Bertini; gRc; Macaulay2
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Anderson T. W. , ’Estimation of covariance matrices which are linear combinations or whose inverses are linear combinations of given matrices’, Essays in probability and statistics (eds R. C. Bose , I. M. Chakravarti , P. C. Mahalanobis , C. R. Rao and K. J. C. Smith ; University of North Carolina Press, Chapel Hill, NC, 1970) 1–24.
[2] Aomoto, On the structure of integrals of power product of linear functions, Sci. Papers College Gen. Ed. Univ. Tokyo 27 pp 49– (1977)
[3] Bates D. , Hauenstein J. , Sommese A. Wampler C. , Numerically solving polynomial systems with Bertini, Software, Environments and Tools 25 (SIAM, Philadelphia, 2013). · Zbl 1295.65057
[4] Blekherman G. , Parrilo P. Thomas R. , Semidefinite optimization and convex algebraic geometry, MOS-SIAM Series on Optimization 13 (SIAM, Philadelphia, 2013). · Zbl 1260.90006
[5] DOI: 10.1007/s11590-013-0694-6 · Zbl 1333.90125 · doi:10.1007/s11590-013-0694-6
[6] Brown L. D. , Fundamentals of statistical exponential families with applications in statistical decision theory (Institute of Mathematical Statistics, Hayward, CA, 1986). · Zbl 0685.62002
[7] DOI: 10.1137/090765328 · Zbl 1252.68134 · doi:10.1137/090765328
[8] Choquet, Deux exemples classiques de représentation intégrale, Enseign. Math. 15 pp 63– (1969)
[9] Croitoru D. , ’Mixed volumes of hypersimplices, root systems and shifted young tableaux’, PhD Dissertation, Massachusetts Institute of Technology, 2010.
[10] DOI: 10.1007/s10208-012-9127-7 · Zbl 1254.90108 · doi:10.1007/s10208-012-9127-7
[11] DOI: 10.2307/2528966 · doi:10.2307/2528966
[12] Drton, Algebraic statistical models, Statist. Sinica 17 pp 1273– (2007) · Zbl 1132.62003
[13] Fulton W. , Introduction to toric varieties (Princeton University Press, Princeton, NJ, 1993). · Zbl 0813.14039 · doi:10.1515/9781400882526
[14] DOI: 10.1007/BF02395740 · Zbl 0045.20202 · doi:10.1007/BF02395740
[15] Gray, Toeplitz and circulant matrices: a review, Found. Trends Commun. Inform. Theory 2 pp 155– (2006) · Zbl 1143.15305 · doi:10.1561/0100000006
[16] Grayson D. R. Stillman M. E. , ’Macaulay2, a software system for research in algebraic geometry’, Preprint, http://www.math.uiuc.edu/Macaulay2/ .
[17] DOI: 10.1287/moor.21.4.860 · Zbl 0867.90090 · doi:10.1287/moor.21.4.860
[18] DOI: 10.1287/moor.22.2.350 · Zbl 0883.90099 · doi:10.1287/moor.22.2.350
[19] DOI: 10.1016/j.jmva.2013.03.011 · Zbl 1282.33010 · doi:10.1016/j.jmva.2013.03.011
[20] Helmke, Bezoutians, Linear Algebra Appl. 122 pp 1039– (1989) · doi:10.1016/0024-3795(89)90684-8
[21] DOI: 10.1023/A:1021852421716 · Zbl 1012.05046 · doi:10.1023/A:1021852421716
[22] Højsgaard, Graphical Gaussian models with edge and vertex symmetries, J. R. Stat. Soc.: Ser. B (Stat. Methodol.) 5 pp 1005– (2008) · doi:10.1111/j.1467-9868.2008.00666.x
[23] Huh J. Sturmfels B. , ’Likelihood geometry’, Combinatorial algebraic geometry (eds A. Conca et al.), Lecture Notes in Mathematics 2108 (Springer, Berlin, 2014) 63–117. · Zbl 1328.14004
[24] DOI: 10.1214/aos/1176350707 · Zbl 0653.62042 · doi:10.1214/aos/1176350707
[25] Kummer
[26] Kummer, Hyperbolic polynomials, interlacers, and sums of squares, Math. Program., Ser. B (2012) · Zbl 1349.14181
[27] Lauritzen S. L. , Graphical models (Oxford University Press, Oxford, 1996). · Zbl 0907.62001
[28] Miller E. Sturmfels B. , Combinatorial commutative algebra, Graduate Texts in Mathematics (Springer, New York, 2004). · Zbl 1090.13001
[29] Pachter L. Sturmfels B. , Algebraic statistics for computational biology (Cambridge University Press, Cambridge, 2005). · Zbl 1108.62118 · doi:10.1017/CBO9780511610684
[30] Postnikov, Permutohedra, associahedra, and beyond, Int. Math. Res. Not. 6 pp 1026– (2009)
[31] DOI: 10.1016/j.ijar.2011.01.013 · Zbl 1214.62013 · doi:10.1016/j.ijar.2011.01.013
[32] Renegar J. , A mathematical view of interior-point methods in convex optimization, MOS-SIAM Series on Optimization (SIAM, Philadelphia, PA, 2001). · Zbl 0986.90075 · doi:10.1137/1.9780898718812
[33] DOI: 10.1112/S002557931100132X · Zbl 1315.52001 · doi:10.1112/S002557931100132X
[34] DOI: 10.1007/s11511-014-0121-6 · Zbl 1304.05074 · doi:10.1007/s11511-014-0121-6
[35] Sturmfels B. , Algorithms in invariant theory (Springer, Vienna and New York, 1993). · Zbl 0802.13002 · doi:10.1007/978-3-7091-4368-1
[36] DOI: 10.1007/s10463-010-0295-4 · Zbl 1440.62255 · doi:10.1007/s10463-010-0295-4
[37] Tsukerman
[38] DOI: 10.1007/BF01077982 · Zbl 0625.33006 · doi:10.1007/BF01077982
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.