The correlation space of Gaussian latent tree models and model selection without fitting

Shiers, Nathaniel; Zwiernik, Piotr; Aston, John A. D.; Smith, Jim Q.

doi:10.48550/arxiv.1508.00436

Cited by 3 publications

(10 citation statements)

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“…However, we show that to obtain the maximum rate region to synthesize the output, one may minimize I(X; Y), which in turn will be equivalent to maximizing the conditional mutual information I(X; B|Y), hence, showing the maximum amount of lost sign information. In such settings, we show that the input B and the output X are independent, by which we provide another reason on why previous learning approaches [1,3] are incapable of inferring the sign information.…”

Section: Introductionmentioning

confidence: 76%

“…In particular, such an approach leads us into two equivalent cases with sign inputs b (1) or −b (1) , with the latter obtained by flipping the signs of all correlations ρ yx i , i ∈ {1, 2, 3}. From [3], we know for the channel shown in Figure 1, we should have ρ x 1 x 2 ρ x 1 x 3 ρ x 2 x 3 > 0. Hence, there are totally two cases for ρ x i x j , i = j, i, j ∈ {1, 2, 3} based on such constraint; either all of them are positive, or two of them are negative and the third one is positive.…”

Section: Studying the Properties Of Sign Information Vector Bmentioning

confidence: 99%

“…Also, as we may observe from Theorem 2, given X i we may end up with several options for X j i and X k i . However, it can be shown that in a subspace of correlations corresponding to latent Gaussian trees [3], all those distinct options result in a same value for the term…”

Section: Maximum Achievable Rate Region To Generate the Output Xmentioning

confidence: 99%

“…They introduced a tree metric as the negative log of the absolute value of pairwise correlations to perform the algorithm. Also, Shiers et al, in [3], characterized the correlation space of latent Gaussian trees and showed the necessary and sufficient conditions under which the correlation space represents a particular latent Gaussian tree. Note that the RG algorithm can be directly related to correlation space of latent Gaussian trees in a sense that it recursively checks certain constraints on correlations to converge to a latent tree with true parameters.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Synthesis of Gaussian trees with correlation sign ambiguity: An information theoretic approach

Moharrer

Wei

Amariucai

et al. 2016

2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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In latent Gaussian trees the pairwise correlation signs between the variables are intrinsically unrecoverable. Such information is vital since it completely determines the direction in which two variables are associated. In this work, we resort to information theoretical approaches to achieve two fundamental goals: First, we quantify the amount of information loss due to unrecoverable sign information. Second, we show the importance of such information in determining the maximum achievable rate region, in which the observed output vector can be synthesized, given its probability density function. In particular, we model the graphical model as a communication channel and propose a new layered encoding framework to synthesize observed data using upper layer Gaussian inputs and independent Bernoulli correlation sign inputs from each layer. We find the achievable rate region for the rate tuples of multi-layer latent Gaussian messages to synthesize the desired observables. 1 A. Moharrer, and S. Wei are with the school

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Section: Introductionmentioning

confidence: 76%

Section: Studying the Properties Of Sign Information Vector Bmentioning

confidence: 99%

Section: Maximum Achievable Rate Region To Generate the Output Xmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Synthesis of Gaussian trees with correlation sign ambiguity: An information theoretic approach

Moharrer

Wei

Amariucai

et al. 2016

2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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show abstract

“…They have diverse applications in social networks, biology, and economics [1], [2], to name a few. Gaussian trees in particular have attracted much attention [2] due to their sparse structures, as well as existing computationally efficient algorithms in learning the underlying topologies [3], [4]. In this paper we assume that the parameters and structure information of the latent Gaussian tree is provided.…”

Section: Introductionmentioning

confidence: 99%

Successive synthesis of latent Gaussian trees

Moharrer

Wei

Amariucai

et al. 2017

MILCOM 2017 - 2017 IEEE Military Communications Conference (MILCOM)

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A new synthesis scheme is proposed to effectively generate a random vector with prescribed joint density that induces a (latent) Gaussian tree structure. The quality of synthesis is measured by total variation distance between the synthesized and desired statistics. The proposed layered and successive encoding scheme relies on the learned structure of tree to use minimal number of common random variables to synthesize the desired density. We characterize the achievable rate region for the rate tuples of multi-layer latent Gaussian tree, through which the number of bits needed to simulate such Gaussian joint density are determined. The random sources used in our algorithm are the latent variables at the top layer of tree, the additive independent Gaussian noises, and the Bernoulli sign inputs that capture the ambiguity of correlation signs between the variables.

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Phylogenetic Forest Spaces

Lueg¹

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Tag der mündlichen Prüfung: 04. Oktober 2022 v vi A promising construction that covers the example from above and behaves as biological intuition suggests is our recently introduced Wald Space, cf. Garba et al. (2021a). It is based on the characterization of phylogenetic trees as covariance matrices, which is a byproduct of a generalization of the popular biological substitution models that are used to calculate likelihoods for trees given genetic sequence data, so those substitution models are the backbone of phylogenetic tree estimation. In other words, the Wald Space is a space that is consistent with the tree estimation methods that are currently used, up to the generalizations that have been made. More details can be found in Garba et al. (2021a).In this work, we concentrate on the Wald Space purely from the perspective that it is a mathematical structure and thus we try to enable for a be er understanding of the Wald Space. To this end, we introduce the mathematical structures required: metric spaces, Riemannian manifolds, Riemann strati ed spaces, as well as, for our construction essential, various geometries on the manifold of strictly positive de nite symmetric real matrices. Furthermore, we introduce various possible ways to represent the phylogenetic trees and forests that we consider. Having nished the introduction of the more general and known concepts, we brie y introduce the BHV Space. en we de ne and describe the Wald Space, which is a topological strati ed space, and we investigate its topological features. is part is the core of the thesis. Finally, we equip the Wald Space with a geometry that can be chosen to some degree and nd that these spaces are then Riemann strati ed spaces of type (A). Last but not least, we propose some numerical algorithms to calculate geodesics and distances in the Wald Space equipped with a geometry.ose are not tested in this work, but to some extent in Lueg et al. (2021).

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The correlation space of Gaussian latent tree models and model selection without fitting

Cited by 3 publications

References 0 publications

Synthesis of Gaussian trees with correlation sign ambiguity: An information theoretic approach

Synthesis of Gaussian trees with correlation sign ambiguity: An information theoretic approach

Successive synthesis of latent Gaussian trees

Phylogenetic Forest Spaces

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