scholarly journals Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 840 ◽  
Author(s):  
Frédéric Barbaresco
Keyword(s):  
Machine Learning ◽  
Maximum Entropy ◽  
Lie Groups ◽  
Lie Group ◽  
Data Analytics ◽  
Geometric Theory ◽  
Higher Order ◽  
Small Data ◽  
Algebra Structure ◽  
Fisher Metric

We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by Ingarden. This extended model could be used for small data analytics and machine learning on Lie groups. Souriau geometric theory of heat is well adapted to describe density of probability (maximum entropy Gibbs density) of data living on groups or on homogeneous manifolds. For small data analytics (rarified gases, sparse statistical surveys, …), the density of maximum entropy should consider higher order moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by Ingarden. We use a poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of moment mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous systems. The formalism is covariant, i.e., no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures. We also introduce a more synthetic Koszul definition of Fisher Metric, based on the Souriau model, that we name Souriau-Fisher metric. This Lie groups thermodynamics is the bedrock for Lie group machine learning providing a full covariant maximum entropy Gibbs density based on representation theory (symplectic structure of coadjoint orbits for Souriau non-equivariant model associated to a class of co-homology).

scholarly journals Higher Order Geometric Theory of Information and Heat based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures

2018 ◽  
Author(s):  
Frederic Barbaresco
Keyword(s):  
Maximum Entropy ◽  
Lie Groups ◽  
Data Analytics ◽  
Statistical Physics ◽  
Symplectic Geometry ◽  
Geometric Theory ◽  
Higher Order ◽  
Small Data ◽  
Extended Model ◽  
Algebra Structure

We introduce poly-symplectic extension of Souriau Lie groups Thermodynamics based on higher-order model of statistical physics introduced by R.S. Ingarden. This extended model could be used for small data analytics and Machine Learning on Lie groups. Souriau Geometric Theory of Heat is well adapted to describe density of probability (Maximum Entropy Gibbs density) of data living on groups or on homogeneous manifolds. For Small Data Analytics (Rarified Gases , sparse statistical survey,…), density of maximum entropy should consider Higher Order Moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by R.S. Ingarden. We use Poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of momentum mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous systems. The formalism is covariant, i.e. no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures.

scholarly journals The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yu Liu ◽  
Jianfeng Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Riesz Transform ◽  
Integral Operators ◽  
Higher Order ◽  
Reverse Hölder Class ◽  
Stratified Lie Group ◽  
Homogeneous Dimension ◽  
Stratified Lie Groups ◽  
Reverse Hölder

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

Some Results in the Connective K-Theory of Lie Groups

1988 ◽  
Vol 31 (2) ◽  
pp. 194-199
Author(s):  
L. Magalhães
Keyword(s):  
Hopf Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
K Theory ◽  
Torsion Free ◽  
Connected Lie Group

AbstractIn this paper we give a description of:(1) the Hopf algebra structure of k*(G; L) when G is a compact, connected Lie group and L is a ring of type Q(P) so that H*(G; L) is torsion free;(2) the algebra structure of k*(G2; L) for L = Z2 or Z.

Machine Learning and Data Analytics in Pervasive Health

2018 ◽  
Vol 57 (04) ◽  
pp. 194-196
Author(s):  
Nuria Oliver ◽  
Michael Marschollek ◽  
Oscar Mayora
Keyword(s):  
Machine Learning ◽  
Medical Informatics ◽  
Data Analytics ◽  
Peer Review Process ◽  
Personalised Medicine ◽  
Physiological Data ◽  
European Federation ◽  
Small Data ◽  
Health Technologies ◽  
Pervasive Health

Summary Introduction: This accompanying editorial provides a brief introduction to this focus theme, focused on “Machine Learning and Data Analytics in Pervasive Health”. Objective: The innovative use of machine learning technologies combining small and big data analytics will support a better provisioning of healthcare to citizens. This focus theme aims to present contributions at the crossroads of pervasive health technologies and data analytics as key enablers for achieving personalised medicine for diagnosis and treatment purposes. Methods: A call for paper was announced to all participants of the “11th International Conference on Pervasive Computing Technologies for Healthcare”, to different working groups of the International Medical Informatics Association (IMIA) and European Federation of Medical Informatics (EFMI) and was published in June 2017 on the website of Methods of Information in Medicine. A peer review process was conducted to select the papers for this focus theme. Results: Four papers were selected to be included in this focus theme. The paper topics cover a broad range of machine learning and data analytics applications in healthcare including detection of injurious subtypes of patient-ventilator asynchrony, early detection of cognitive impairment, effective use of small data sets for estimating the performance of radiotherapy in bladder cancer treatment, and the use negation detection in and information extraction from unstructured medical texts. Conclusions: The use of machine learning and data analytics technologies in healthcare is facing a renewed impulse due to the availability of large amounts and new sources of human behavioral and physiological data, such as that captured by mobile and pervasive devices traditionally considered as nonmainstream for healthcare provision and management.

scholarly journals Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

2016 ◽  
Author(s):  
Frédéric Barbaresco
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Statistical Physics ◽  
Geometric Theory ◽  
Adjoint Action ◽  
Information Geometry ◽  
Affine Group ◽  
Exponential Families ◽  
Natural Exponential Families ◽  
Fisher Metric

We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of Information Geometry for the action of an affine Group for exponentiel families, and provide some illustration of use cases for multivariate Gaussian densities. Information Geometry is presented in the context of seminal work of Fréchet and his Clairait-Legendre equation. Souriau model of Statistical Physics is validated as compatible with Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

scholarly journals Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

2016 ◽  
Author(s):  
Frédéric Barbaresco
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Statistical Physics ◽  
Geometric Theory ◽  
Adjoint Action ◽  
Information Geometry ◽  
Affine Group ◽  
Exponential Families ◽  
Natural Exponential Families ◽  
Fisher Metric

We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of Information Geometry for the action of an affine Group for exponentiel families, and provide some illustrations of use cases for multivariate Gaussian densities. Information Geometry is presented in the context of seminal work of Fréchet and his Clairaut-Legendre equation. Souriau model of Statistical Physics is validated as compatible with Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

scholarly journals Higher-Order Differential Operators on a Lie Group and Quantization

1997 ◽  
Vol 12 (01) ◽  
pp. 3-11 ◽  
Author(s):  
V. Aldaya ◽  
J. Guerrero ◽  
G. Marmo
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Adjoint Method ◽  
Differential Operators ◽  
Higher Order ◽  
Group Approach ◽  
Representations Of Lie Groups

This talk is devoted mainly to the concept of higher-order polarization on a group, which is introduced in the framework of a Group Approach to Quantization, as a powerful tool to guarantee the irreducibility of quantization and/or representations of Lie groups in those anomalous cases where the Kostant–Kirilov co-adjoint method or the Borel–Weyl–Bott representation algorithm do not succeed.

scholarly journals Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 498 ◽  
Author(s):  
Frédéric Barbaresco ◽  
François Gay-Balmaz
Keyword(s):  
Machine Learning ◽  
Quantum Information ◽  
Statistical Mechanics ◽  
Lie Group ◽  
Information Geometry ◽  
Symplectic Integrators ◽  
Coadjoint Orbits ◽  
Probability Densities ◽  
Fisher Metric

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

The mod 2 homology of the space of loops on the exceptional Lie group

1989 ◽  
Vol 112 (3-4) ◽  
pp. 187-202 ◽  
Author(s):  
Akira Kono ◽  
Kazumuto Kozima
Keyword(s):  
Hopf Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Steenrod Algebra ◽  
Algebra Structure ◽  
Spectral Sequences ◽  
Hopf Algebra Structure ◽  
Simple Lie Groups ◽  
Simply Connected

SynopsisThe Hopf algebra structure of H*(ΩG, F2) and the action of the dual Steenrod algebra are completely and explicitly determined when G isone of the connected, simply connected, exceptional, simple Lie groups. The approach is homological, using connected coverings and spectral sequences.

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