Information Geometry
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Entropy-regularized 2-Wasserstein distance between Gaussian measures
Information Geometry - Tập 5 - Trang 289-323 - 2021
Gaussian distributions are plentiful in applications dealing in uncertainty
quantification and diffusivity. They furthermore stand as important special
cases for frameworks providing geometries for probability measures, as the
resulting geometry on Gaussians is often expressible in closed-form under the
frameworks. In this work, we study the Gaussian geometry under the
entropy-regularized 2-Wasser...
Information geometry of warped product spaces
Information Geometry - Tập 6 - Trang 127-155 - 2022
Information geometry is an important tool to study statistical models. There are
some important examples in statistical models which are regarded as warped
products. In this paper, we study information geometry of warped products. We
consider the case where the warped product and its fiber space are equipped with
dually flat connections and, in the particular case of a cone, characterize the
conne...
Anomaly detection in the probability simplex under different geometries
Information Geometry - Tập 6 - Trang 385-412 - 2023
An open problem in data science is that of anomaly detection. Anomalies are
instances that do not maintain a certain property that is present in the
remaining observations in a dataset. Several anomaly detection algorithms exist,
since the process itself is ill-posed mainly because the criteria that separates
common or expected vectors from anomalies are not unique. In the most extreme
case, data ...
Geometry and applied statistics
Information Geometry - Tập 7 - Trang 211-227 - 2022
We take a very high level overview of the relationship between Geometry and
Applied Statistics 50 years from the birth of Information Geometry. From that
date we look both backwards and forwards. We show that Geometry has always been
part of the statistician’s toolbox and how it played a vital role in the
evolution of Statistics in the last 50 years.
The Bonnet theorem for statistical manifolds
Information Geometry - Tập 4 - Trang 363-376 - 2021
We prove the Bonnet theorem for statistical manifolds, which states that if a
statistical manifold admits tensors satisfying the Gauss–Codazzi–Ricci
equations, then it is locally embeddable to a flat statistical manifold (or a
Hessian manifold). The proof is based on the notion of statistical embedding to
the product of a vector space and its dual space introduced by Lauritzen. As
another applicat...
Coarse geometric kernels for networks embedding
Information Geometry - Tập 6 - Trang 157-169 - 2023
We develop embedding kernels based on the Forman–Ricci curvature and intertwined
Bochner–Laplacian and employ them for the detection of the coarse structure of
networks, as well as for network visualization with applications to
support-vector machines (SVMs).
Transport information geometry: Riemannian calculus on probability simplex
Information Geometry - Tập 5 - Trang 161-207 - 2021
We formulate the Riemannian calculus of the probability set embedded with
$$L^2$$ -Wasserstein metric. This is an initial work of transport information
geometry. Our investigation starts with the probability simplex (probability
manifold) supported on vertices of a finite graph. The main idea is to embed the
probability manifold as a submanifold of the positive measure space with a
weighted graph ...
Parametric models and information geometry on W*-algebras
Information Geometry - Tập 7 - Trang 329-354 - 2023
We introduce the notion of smooth parametric model of normal positive linear
functionals on possibly infinite-dimensional $$W^{\star }$$ -algebras
generalizing the notions of parametric models used in classical and quantum
information geometry. We then use the Jordan product naturally available in this
context in order to define a Riemannian metric tensor on parametric models
satsfying suitable re...
G-dual Teleparallel Connections in Information Geometry
Information Geometry - Tập 7 - Trang 587-608 - 2023
Given a real, finite-dimensional, smooth parallelizable Riemannian manifold
$$(\mathcal {N},G)$$ endowed with a teleparallel connection $$\nabla $$
determined by a choice of a global basis of vector fields on $$\mathcal {N}$$ ,
we show that the G-dual connection $$\nabla ^{*}$$ of $$\nabla $$ in the sense
of Information Geometry must be the teleparallel connection determined by the
basis of G-grad...
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