Poly-symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics

2017 ◽  
pp. 432-441
Author(s):  
Frederic Barbaresco
Keyword(s):  
Lie Groups ◽  
Data Analytics ◽  
Higher Order ◽  
Small Data

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.

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

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).

Uncertainty and grey data analytics

2019 ◽  
Vol 2 (2) ◽  
pp. 73-86 ◽  
Author(s):  
Yingjie Yang ◽  
Sifeng Liu ◽  
Naiming Xie
Keyword(s):  
Social Life ◽  
Data Analytics ◽  
Radical Change ◽  
Point Of View ◽  
Small Data ◽  
Content Type ◽  
Proposed Model ◽  
Uncertainty Models ◽  
First Time ◽  
Practical Implications

Purpose The purpose of this paper is to propose a framework for data analytics where everything is grey in nature and the associated uncertainty is considered as an essential part in data collection, profiling, imputation, analysis and decision making. Design/methodology/approach A comparative study is conducted between the available uncertainty models and the feasibility of grey systems is highlighted. Furthermore, a general framework for the integration of grey systems and grey sets into data analytics is proposed. Findings Grey systems and grey sets are useful not only for small data, but also big data as well. It is complementary to other models and can play a significant role in data analytics. Research limitations/implications The proposed framework brings a radical change in data analytics. It may bring a fundamental change in our way to deal with uncertainties. Practical implications The proposed model has the potential to avoid the mistake from a misleading data imputation. Social implications The proposed model takes the philosophy of grey systems in recognising the limitation of our knowledge which has significant implications in our way to deal with our social life and relations. Originality/value This is the first time that the whole data analytics is considered from the point of view of grey systems.

Small Data, Big Data, and Data Analytics

2018 ◽  
pp. 73-92
Author(s):  
Paul Cerrato ◽  
John Halamka
Keyword(s):  
Big Data ◽  
Data Analytics ◽  
Small Data

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 Higher-order variational problems on lie groups and optimal control applications

2014 ◽  
Vol 6 (4) ◽  
pp. 451-478 ◽  
Author(s):  
Leonardo Colombo ◽  
◽  
David Martín de Diego ◽  
Keyword(s):  
Optimal Control ◽  
Lie Groups ◽  
Variational Problems ◽  
Higher Order ◽  
Control Applications

ON THE EQUATIONS GOVERNING THE FLOW OF MECHANICALLY INCOMPRESSIBLE, BUT THERMALLY EXPANSIBLE, VISCOUS FLUIDS

2008 ◽  
Vol 18 (06) ◽  
pp. 813-857 ◽  
Author(s):  
ANGIOLO FARINA ◽  
ANTONIO FASANO ◽  
ANDRO MIKELIĆ
Keyword(s):  
Stationary Flow ◽  
Boussinesq Approximation ◽  
Momentum Equation ◽  
Higher Order ◽  
Order Correction ◽  
Boussinesq System ◽  
Small Data ◽  
Heat Conducting ◽  
Existence Of A Solution ◽  
Advection Diffusion Equation

In this paper, the stationary flow of a heat conducting viscous fluid, which is mechanically incompressible but thermally expansible is studied. The flow takes place in a bounded domain and the discharge is prescribed. The thermodynamical modeling of this situation is discussed first. Then the stationary model with zero Eckert number and prove existence of a solution is studied. Using these results, the Oberbeck–Boussinesq system obtained in the zero expansivity limit is proved. Next, uniqueness for small data and the regularity of the weak solutions is proved. For the unique regular solution the higher order corrections for Boussinesq' approximation and we constructed the error estimate with respect to the thermal expansivity coefficient is proved. The next order correction in this limit is an Oseen type momentum equation coupled with a linear advection/diffusion equation for the temperature. Such higher order correction seems to be new in the literature.

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