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Dear Colleagues,

One can expect that the development of statistical geometry will be a major trend in the next decade. Due to numerous applications in data analysis, artificial intelligence and signal processing, along with a strong potential for new theoretical results, this emerging axis of research is of increasing interest among both the geometry and statistics communities.

This Special Issue aims at publishing papers of high-quality related to information geometry, geometric statistics, topology for learning and data analysis.

Some of the scientific topics of interest may include, but are not limited to: probabilities on manifolds, shape space, optimal transport, statistical manifolds, geometry of graphs, topology in data, geometry in artificial intelligence, estimation and sampling in high dimensions.

Submissions describing applications are welcome.

Prof. Dr. Stéphane Puechmorel
Guest Editor

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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Keywords

information geometry
topological data analysis
hessian geometry
shape space
optimal transport

Published Papers (3 papers)

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Research

10 pages, 2125 KiB

Open AccessArticle

Topology Adaptive Graph Estimation in High Dimensions

by Johannes Lederer and Christian L. Müller

Mathematics 2022, 10(8), 1244; https://doi.org/10.3390/math10081244 - 10 Apr 2022

Viewed by 1278

Abstract

We introduce Graphical TREX (GTREX), a novel method for graph estimation in high-dimensional Gaussian graphical models. By conducting neighborhood selection with TREX, GTREX avoids tuning parameters and is adaptive to the graph topology. We compared GTREX with standard methods on a new simulation setup that was designed to assess accurately the strengths and shortcomings of different methods. These simulations showed that a neighborhood selection scheme based on Lasso and an optimal (in practice unknown) tuning parameter outperformed other standard methods over a large spectrum of scenarios. Moreover, we show that GTREX can rival this scheme and, therefore, can provide competitive graph estimation without the need for tuning parameter calibration. Full article

(This article belongs to the Special Issue Geometry and Topology in Statistics)

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15 pages, 266 KiB

Open AccessArticle

Lifting Dual Connections with the Riemann Extension

by Stéphane Puechmorel

Mathematics 2020, 8(11), 2079; https://doi.org/10.3390/math8112079 - 21 Nov 2020

Cited by 3 | Viewed by 1646

Abstract

Let

(M, g)

be a Riemannian manifold equipped with a pair of dual connections

(\nabla, \nabla^{*})

. Such a structure is known as a statistical manifold since it was defined in the context of information geometry. [...] Read more.

Let

(M, g)

be a Riemannian manifold equipped with a pair of dual connections

(\nabla, \nabla^{*})

. Such a structure is known as a statistical manifold since it was defined in the context of information geometry. This paper aims at defining the complete lift of such a structure to the cotangent bundle

T^{*} M

using the Riemannian extension of the Levi-Civita connection of M. In the first section, common tensors are associated with pairs of dual connections, emphasizing the cyclic symmetry property of the so-called skewness tensor. In a second section, the complete lift of this tensor is obtained, allowing the definition of dual connections on

T T^{*} M

with respect to the Riemannian extension. This work was motivated by the general problem of finding the projective limit of a sequence of a finite-dimensional statistical manifold. Full article

(This article belongs to the Special Issue Geometry and Topology in Statistics)

13 pages, 323 KiB

Open AccessArticle

Diffeological Statistical Models, the Fisher Metric and Probabilistic Mappings

by Hông Vân Lê

Mathematics 2020, 8(2), 167; https://doi.org/10.3390/math8020167 - 30 Jan 2020

Cited by 3 | Viewed by 1834

Abstract

We introduce the notion of a

C^{k}

-diffeological statistical model, which allows us to apply the theory of diffeological spaces to (possibly singular) statistical models. In particular, we introduce a class of almost 2-integrable

C^{k}

-diffeological statistical models that encompasses all [...] Read more.

We introduce the notion of a

C^{k}

-diffeological statistical model, which allows us to apply the theory of diffeological spaces to (possibly singular) statistical models. In particular, we introduce a class of almost 2-integrable

C^{k}

-diffeological statistical models that encompasses all known statistical models for which the Fisher metric is defined. This class contains a statistical model which does not appear in the Ay–Jost–Lê–Schwachhöfer theory of parametrized measure models. Then, we show that, for any positive integer k , the class of almost 2-integrable

C^{k}

-diffeological statistical models is preserved under probabilistic mappings. Furthermore, the monotonicity theorem for the Fisher metric also holds for this class. As a consequence, the Fisher metric on an almost 2-integrable

C^{k}

-diffeological statistical model

P \subset P (X)

is preserved under any probabilistic mapping

T : X ⇝ Y

that is sufficient w.r.t. P. Finally, we extend the Cramér–Rao inequality to the class of 2-integrable

C^{k}

-diffeological statistical models. Full article

(This article belongs to the Special Issue Geometry and Topology in Statistics)

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