Exponential varieties

Michałek, Mateusz; Sturmfels, Bernd; Uhler, Caroline; Zwiernik, Piotr

doi:10.1112/plms/pdv066

Cited by 24 publications

(31 citation statements)

References 34 publications

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“…On the one hand, finite discrete exponential families have been shown to correspond to toric varieties [9]. On the other hand, some continuous exponential families have been linked to what are called exponential varieties [17]. The present paper now relates some infinite discrete exponential families to abelian varieties.…”

Section: Introductionmentioning

confidence: 90%

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Discrete Gaussian Distributions via Theta Functions

Agostini

Améndola

2019

SIAM J. Appl. Algebra Geometry

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We study a discrete analogue of the classical multivariate Gaussian distribution. It is supported on the integer lattice and is parametrized by the Riemann theta function. Over the reals, the discrete Gaussian is characterized by the property of maximizing entropy, just as its continuous counterpart. We capitalize on the theta function representation to derive statistical properties. Throughout, we exhibit strong connections to the study of abelian varieties in algebraic geometry.

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Section: Introductionmentioning

confidence: 90%

“…Proof. From classical theory of minimal regular exponential families [6,17], there is a bijection between the space of canonical parameters {η | A(η) < ∞} and the space of sufficient statistics conv(T (x)|x ∈ X ), and it is given by the gradient of the log-partition function ∇A(η).…”

Section: The Discrete Gaussian Distribution and The Riemann Theta Funmentioning

confidence: 99%

Discrete Gaussian Distributions via Theta Functions

Agostini

Améndola

2019

SIAM J. Appl. Algebra Geometry

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“…One of their motivations is connections with the cohomology of the complement of a hyperplane arrangement. These varieties also appear in the algebraic study of interior point methods for linear programming [4] and entropy maximization for log-linear models in statistics [9].…”

Section: 4mentioning

confidence: 99%

Semi-inverted linear spaces and an analogue of the broken circuit complex

Scholten¹,

Vinzant²

2019

Algebraic Combinatorics

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The image of a linear space under inversion of some coordinates is an affine variety whose structure is governed by an underlying hyperplane arrangement. In this paper, we generalize work by Proudfoot and Speyer to show that circuit polynomials form a universal Gröbner basis for the ideal of polynomials vanishing on this variety. The proof relies on degenerations to the Stanley-Reisner ideal of a simplicial complex determined by the underlying matroid, which is closely related to the external activity complex defined by Ardila and Boocher. If the linear space is real, then the semi-inverted linear space is also an example of a hyperbolic variety, meaning that all of its intersection points with a large family of linear spaces are real.

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“…In the language of commutative algebra, it is the real Zariski closure of the exponential family model, cf. [29]. It is a notable example of toric variety.…”

Section: {+1 −1}mentioning

confidence: 99%

Natural Gradient Flow in the Mixture Geometry of a Discrete Exponential Family

Malagò

Pistone

2015

Entropy

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In this paper, we study Amari's natural gradient flows of real functions defined on the densities belonging to an exponential family on a finite sample space. Our main example is the minimization of the expected value of a real function defined on the sample space. In such a case, the natural gradient flow converges to densities with reduced support that belong to the border of the exponential family. We have suggested in previous works to use the natural gradient evaluated in the mixture geometry. Here, we show that in some cases, the differential equation can be extended to a bigger domain in such a way that the densities at the border of the exponential family are actually internal points in the extended problem. The extension is based on the algebraic concept of an exponential variety. We study in full detail a toy example and obtain positive partial results in the important case of a binary sample space.

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Exponential varieties

Cited by 24 publications

References 34 publications

Discrete Gaussian Distributions via Theta Functions

Discrete Gaussian Distributions via Theta Functions

Semi-inverted linear spaces and an analogue of the broken circuit complex

Natural Gradient Flow in the Mixture Geometry of a Discrete Exponential Family

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