Mechanical Systems: Symmetries and Reduction

Marsden, Jerrold E.; Raţiu, Tudor S.

doi:10.1007/978-0-387-30440-3_326

Cited by 2 publications

(1 citation statement)

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“…Using left-invariant vector fields, one can compute explicitly the LC connection, allowing to rewrite the geodesic equation. This fundamental idea, known as Euler-Poincaré reduction, is that the geodesic equation can be expressed entirely in the Lie algebra thanks to the symmetry of left-invariance (Marsden and Ratiu, 2009), alleviating the burden of coordinate charts.…”

Section: Rationalementioning

confidence: 99%

Introduction to Riemannian Geometry and Geometric Statistics: From Basic Theory to Implementation with Geomstats

Guigui¹,

Miolane

Pennec³

2023

FNT in Machine Learning

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As data is a predominant resource in applications, Riemannian geometry is a natural framework to model and unify complex nonlinear sources of data. However, the development of computational tools from the basic theory of Riemannian geometry is laborious. The work presented here forms one of the main contributions to the open-source project geomstats, that consists of a Python package providing efficient implementations of the concepts of Riemannian geometry and geometric statistics, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions. The goal of this monograph is two-fold. First, we aim at giving a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. The second goal is to demonstrate how these concepts are implemented in Geomstats, explaining the choices that were made and the conventions chosen. The general concepts are exposed and specific examples are detailed along the text. The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. We exemplify this with an introduction to geometric statistics.

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Section: Rationalementioning

confidence: 99%