Tropical Gaussians: a brief survey

Tran, Ngoc Mai

doi:10.2140/astat.2020.11.155

Cited by 7 publications

(6 citation statements)

References 78 publications

(94 reference statements)

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“…The theory of optimal transport and the study of associated density manifolds provide a natural base to construct heat equations with respect to the tropical metric. This provides an important potential to defining non-uniform probability distributions on the tropical projective torus: classically, the solution to the heat equation gives rise to the Gaussian distribution, thus, a solution to the tropical heat equation is a candidate for a tropical Gaussian distribution on the tropical projective torus [11,46]. The dynamic setting of optimal transport with the tropical ground metric introduced in this paper also provides a foundation to studying the displacement convexity and Ricci curvature tensor on the tropical projective torus.…”

Section: Discussionmentioning

confidence: 99%

Tropical optimal transport and Wasserstein distances

Lee

Lin

et al. 2021

Info. Geo.

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We study the problem of optimal transport in tropical geometry and define the Wasserstein-p distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric—a combinatorial metric that has been used to study of the tropical geometric space of phylogenetic trees—as the ground metric and study the cases of $$p=1,2$$ p = 1 , 2 in detail. The case of $$p=1$$ p = 1 gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of $$p=2$$ p = 2 gives a form for Fréchet means and a general inner product structure. Our results also provide theoretical foundations for geometric insight a statistical framework in a tropical geometric setting. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport in tropical geometry. Several numerical examples are provided.

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Section: Discussionmentioning

confidence: 99%

Tropical optimal transport and Wasserstein distances

Lee

Lin

et al. 2021

Info. Geo.

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“…Then the pdf of a classical Gaussian distribution is where and is the transpose of a vector . When , the tropical Laplacian distribution in ( 6 ) is tropicalization of the left hand side in ( 7 ) where is to the tropical identity matrix Tran ( 2020 ) nicely surveys the many different definitions of tropical Gaussian distributions. Since the space of ultrametrics is a tropical linear space (Speyer and Sturmfels 2009 ), it is natural to use tropical “linear algebra” for the definition of tropical “Gaussian” distribution defined in ( 6 ) in this research.…”

Section: Methodsmentioning

confidence: 99%

Tropical Logistic Regression Model on Space of Phylogenetic Trees

Aliatimis,

Yoshida,

Boyacı

et al. 2024

Bull Math Biol

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Classification of gene trees is an important task both in the analysis of multi-locus phylogenetic data, and assessment of the convergence of Markov Chain Monte Carlo (MCMC) analyses used in Bayesian phylogenetic tree reconstruction. The logistic regression model is one of the most popular classification models in statistical learning, thanks to its computational speed and interpretability. However, it is not appropriate to directly apply the standard logistic regression model to a set of phylogenetic trees, as the space of phylogenetic trees is non-Euclidean and thus contradicts the standard assumptions on covariates. It is well-known in tropical geometry and phylogenetics that the space of phylogenetic trees is a tropical linear space in terms of the max-plus algebra. Therefore, in this paper, we propose an analogue approach of the logistic regression model in the setting of tropical geometry. Our proposed method outperforms classical logistic regression in terms of Area under the ROC Curve in numerical examples, including with data generated by the multi-species coalescent model. Theoretical properties such as statistical consistency have been proved and generalization error rates have been derived. Finally, our classification algorithm is proposed as an MCMC convergence criterion for . Unlike the convergence metric used by which is only dependent on tree topologies, our method is sensitive to branch lengths and therefore provides a more robust metric for convergence. In a test case, it is illustrated that the tropical logistic regression can differentiate between two independently run MCMC chains, even when the standard metric cannot.

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“…One motivation for this paper is the search for a suitable definition of tropical Gaussian measures [26]. Tropical stochastics has been an active research area in recent years and has diverse applications from phylogenetics [20; 29] to game theory [2] and economics [4; 27].…”

Section: The Collection Of Entropy Values (H I ( )) I ⊂[D] Satisfies ...mentioning

confidence: 99%

The Gaussian entropy map in valued fields

Maazouz

2022

Alg. Stat.

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The entropy map for multivariate real valued Gaussian distributions is the map that sends a positive definite matrix to the sequence of logarithms of its principal minors (log(det I )) I . We exhibit the analog of this map in the nonarchimedean local fields setting, like the field of p-adic numbers for example. As in the real case, the image of this map lies in the supermodular cone. Moreover, given a multivariate Gaussian measure on a local field, its image under the entropy map determines its pushforward under valuation. In general, this map can be defined for nonarchimedean valued fields whose valuation group is an additive subgroup of the real line, and it remains supermodular. We also explicitly compute the image of this map in dimension 3.

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Tropical Gaussians: a brief survey

Abstract: We review the existing analogues of the Gaussian measure in the tropical semiring and outline various research directions.

Cited by 7 publications

References 78 publications

Tropical optimal transport and Wasserstein distances

Tropical optimal transport and Wasserstein distances

Tropical Logistic Regression Model on Space of Phylogenetic Trees

The Gaussian entropy map in valued fields

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