The Experts below are selected from a list of 22146 Experts worldwide ranked by ideXlab platform
George Ruppeiner - One of the best experts on this subject based on the ideXlab platform.
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Riemannian Geometry in thermodynamic fluctuation theory
Reviews of Modern Physics, 1995Co-Authors: George RuppeinerAbstract:Although thermodynamic fluctuation theory originated from statistical mechanics, it may be put on a completely thermodynamic basis, in no essential need of any microscopic foundation. This review views the theory from the macroscopic perspective, emphasizing, in particular, notions of covariance and consistency, expressed naturally using the language of Riemannian Geometry. Coupled with these concepts is an extension of the basic structure of thermodynamic fluctuation theory beyond the classical one of a subsystem in contact with an infinite uniform reservoir. Used here is a hierarchy of concentric subsystems, each of which samples only the thermodynamic state of the subsystem immediately larger than it. The result is a covariant thermodynamic fluctuation theory which is plausible beyond the standard second-order entropy expansion. It includes the conservation laws and is mathematically consistent when applied to fluctuations inside subsystems. Tests on known models show improvements. Perhaps most significantly, the covariant theory offers a qualitatively new tool for the study of fluctuation phenomena: the Riemannian thermodynamic curvature. The thermodynamic curvature gives, for any given thermodynamic state, a lower bound for the length scale where the classical thermodynamic fluctuation theory based on a uniform environment could conceivably hold. Straightforward computation near the critical point reveals that themore » curvature equals the correlation volume, a physically appealing finding. The combination of the interpretation of curvature with a well-known proportionality between the free energy and the inverse of the correlation volume yields a purely thermodynamic theory of the critical point. The scaled equation of state follows from the values of the critical exponents. The thermodynamic Riemannian metric may be put into the broader context of information theory.« less
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Riemannian Geometry in thermodynamic fluctuation theory
Reviews of Modern Physics, 1995Co-Authors: George RuppeinerAbstract:Although thermodynamic fluctuation theory originated from statistical mechanics, it may be put on a completely thermodynamic basis, in no essential need of any microscopic foundation. This review views the theory from the macroscopic perspective, emphasizing, in particular, notions of covariance and consistency, expressed naturally using the language of Riemannian Geometry. Coupled with these concepts is an extension of the basic structure of thermodynamic fluctuation theory beyond the classical one of a subsystem in contact with an infinite uniform reservoir. Used here is a hierarchy of concentric subsystems, each of which samples only the thermodynamic state of the subsystem immediately larger than it. The result is a covariant thermodynamic fluctuation theory which is plausible beyond the standard second-order entropy expansion. It includes the conservation laws and is mathematically consistent when applied to fluctuations inside subsystems. Tests on known models show improvements. Perhaps most significantly, the covariant theory offers a qualitatively new tool for the study of fluctuation phenomena: the Riemannian thermodynamic curvature. The thermodynamic curvature gives, for any given thermodynamic state, a lower bound for the length scale where the classical thermodynamic fluctuation theory based on a uniform environment could conceivably hold. Straightforward computation near the critical point reveals that the curvature equals the correlation volume, a physically appealing finding. The combination of the interpretation of curvature with a well-known proportionality between the free energy and the inverse of the correlation volume yields a purely thermodynamic theory of the critical point. The scaled equation of state follows from the values of the critical exponents. The thermodynamic Riemannian metric may be put into the broader context of information theory.
Marco Congedo - One of the best experts on this subject based on the ideXlab platform.
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an unsupervised multichannel artifact detection method for sleep eeg based on Riemannian Geometry
Sensors, 2019Co-Authors: Elizaveta Saifutdinova, Daniela Urbaczka Dudysová, Jana Koprivova, Lenka Lhotska, Marco Congedo, Václav GerlaAbstract:In biomedical signal processing, we often face the problem of artifacts that distort the original signals. This concerns also sleep recordings, such as EEG. Artifacts may severely affect or even make impossible visual inspection, as well as automatic processing. Many proposed methods concentrate on certain artifact types. Therefore, artifact-free data are often obtained after sequential application of different methods. Moreover, single-channel approaches must be applied to all channels alternately. The aim of this study is to develop a multichannel artifact detection method for multichannel sleep EEG capable of rejecting different artifact types at once. The inspiration for the study is gained from recent advances in the field of Riemannian Geometry. The method we propose is tested on real datasets. The performance of the proposed method is measured by comparing detection results with the expert labeling as a reference and evaluated against a simpler method based on Riemannian Geometry that has previously been proposed, as well as against the state-of-the-art method FASTER. The obtained results prove the effectiveness of the proposed method.
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transfer learning a Riemannian Geometry framework with applications to brain computer interfaces
IEEE Transactions on Biomedical Engineering, 2018Co-Authors: Paolo Zanini, Marco Congedo, Christian Jutten, Salem Said, Yannick BerthoumieuAbstract:Objective: This paper tackles the problem of transfer learning in the context of electroencephalogram (EEG)-based brain–computer interface (BCI) classification. In particular, the problems of cross-session and cross-subject classification are considered. These problems concern the ability to use data from previous sessions or from a database of past users to calibrate and initialize the classifier, allowing a calibration-less BCI mode of operation. Methods: Data are represented using spatial covariance matrices of the EEG signals, exploiting the recent successful techniques based on the Riemannian Geometry of the manifold of symmetric positive definite (SPD) matrices. Cross-session and cross-subject classification can be difficult, due to the many changes intervening between sessions and between subjects, including physiological, environmental, as well as instrumental changes. Here, we propose to affine transform the covariance matrices of every session/subject in order to center them with respect to a reference covariance matrix, making data from different sessions/subjects comparable. Then, classification is performed both using a standard minimum distance to mean classifier, and through a probabilistic classifier recently developed in the literature, based on a density function (mixture of Riemannian Gaussian distributions) defined on the SPD manifold. Results: The improvements in terms of classification performances achieved by introducing the affine transformation are documented with the analysis of two BCI datasets. Conclusion and significance: Hence, we make, through the affine transformation proposed, data from different sessions and subject comparable, providing a significant improvement in the BCI transfer learning problem.
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Riemannian Geometry for EEG-based brain-computer interfaces; a primer and a review
Brain-Computer Interfaces, 2017Co-Authors: Marco Congedo, Alexandre Barachant, Rajendra BhatiaAbstract:Despite its short history, the use of Riemannian Geometry in brain-computer interface (BCI) decoding is currently attracting increasing attention, due to accumulating documentation of its simplicity, accuracy, robustness and transfer learning capabilities, including the winning score obtained in five recent international predictive modeling BCI data competitions. The Riemannian framework is sharp from a mathematical perspective, yet in practice it is simple, both algorithmically and computationally. This allows the conception of online decoding machines suiting real-world operation in adverse conditions. We provide here a review on the use of Riemannian Geometry for BCI and a primer on the classification frameworks based on it. While the theoretical research on Riemannian Geometry is technical, our aim here is to show the appeal of the framework on an intuitive geometrical ground. In particular, we provide a rationale for its robustness and transfer learning capabilities and we elucidate the link between a simple Riemannian classifier and a state-of-the-art spatial filtering approach. We conclude by reporting details on the construction of data points to be manipulated in the Riemannian framework in the context of BCI and by providing links to available open-source Matlab and Python code libraries for designing BCI decoders.
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Multiclass Brain-Computer Interface Classification by Riemannian Geometry
IEEE Transactions on Biomedical Engineering, 2012Co-Authors: Alexandre Barachant, Marco Congedo, Stephane Bonnet, Christian JuttenAbstract:This paper presents a new classification framework for brain-computer interface (BCI) based on motor imagery. This framework involves the concept of Riemannian Geometry in the manifold of covariance matrices. The main idea is to use spatial covariance matrices as EEG signal descriptors and to rely on Riemannian Geometry to directly classify these matrices using the topology of the manifold of symmetric and positive definite (SPD) matrices. This framework allows to extract the spatial information contained in EEG signals without using spatial filtering. Two methods are proposed and compared with a reference method [multiclass Common Spatial Pattern (CSP) and Linear Discriminant Analysis (LDA)] on the multiclass dataset IIa from the BCI Competition IV. The first method, named minimum distance to Riemannian mean (MDRM), is an implementation of the minimum distance to mean (MDM) classification algorithm using Riemannian distance and Riemannian mean. This simple method shows comparable results with the reference method. The second method, named tangent space LDA (TSLDA), maps the covariance matrices onto the Riemannian tangent space where matrices can be vectorized and treated as Euclidean objects. Then, a variable selection procedure is applied in order to decrease dimensionality and a classification by LDA is performed. This latter method outperforms the reference method increasing the mean classification accuracy from 65.1% to 70.2%.
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Common Spatial Pattern revisited by Riemannian Geometry
2010Co-Authors: Alexandre Barachant, Marco Congedo, Stephane Bonnet, Christian JuttenAbstract:This paper presents a link between the well known Common Spatial Pattern (CSP) algorithm and Riemannian Geometry in the context of Brain Computer Interface (BCI). It will be shown that CSP spatial filtering and Log variance features extraction can be resumed as a computation of a Riemann distance in the space of covariances matrices. This fact yields to highlight several approximations with respect to the space topology. According to these conclusions, we propose an improvement of classical CSP method.
Christian Jutten - One of the best experts on this subject based on the ideXlab platform.
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transfer learning a Riemannian Geometry framework with applications to brain computer interfaces
IEEE Transactions on Biomedical Engineering, 2018Co-Authors: Paolo Zanini, Marco Congedo, Christian Jutten, Salem Said, Yannick BerthoumieuAbstract:Objective: This paper tackles the problem of transfer learning in the context of electroencephalogram (EEG)-based brain–computer interface (BCI) classification. In particular, the problems of cross-session and cross-subject classification are considered. These problems concern the ability to use data from previous sessions or from a database of past users to calibrate and initialize the classifier, allowing a calibration-less BCI mode of operation. Methods: Data are represented using spatial covariance matrices of the EEG signals, exploiting the recent successful techniques based on the Riemannian Geometry of the manifold of symmetric positive definite (SPD) matrices. Cross-session and cross-subject classification can be difficult, due to the many changes intervening between sessions and between subjects, including physiological, environmental, as well as instrumental changes. Here, we propose to affine transform the covariance matrices of every session/subject in order to center them with respect to a reference covariance matrix, making data from different sessions/subjects comparable. Then, classification is performed both using a standard minimum distance to mean classifier, and through a probabilistic classifier recently developed in the literature, based on a density function (mixture of Riemannian Gaussian distributions) defined on the SPD manifold. Results: The improvements in terms of classification performances achieved by introducing the affine transformation are documented with the analysis of two BCI datasets. Conclusion and significance: Hence, we make, through the affine transformation proposed, data from different sessions and subject comparable, providing a significant improvement in the BCI transfer learning problem.
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Multiclass Brain-Computer Interface Classification by Riemannian Geometry
IEEE Transactions on Biomedical Engineering, 2012Co-Authors: Alexandre Barachant, Marco Congedo, Stephane Bonnet, Christian JuttenAbstract:This paper presents a new classification framework for brain-computer interface (BCI) based on motor imagery. This framework involves the concept of Riemannian Geometry in the manifold of covariance matrices. The main idea is to use spatial covariance matrices as EEG signal descriptors and to rely on Riemannian Geometry to directly classify these matrices using the topology of the manifold of symmetric and positive definite (SPD) matrices. This framework allows to extract the spatial information contained in EEG signals without using spatial filtering. Two methods are proposed and compared with a reference method [multiclass Common Spatial Pattern (CSP) and Linear Discriminant Analysis (LDA)] on the multiclass dataset IIa from the BCI Competition IV. The first method, named minimum distance to Riemannian mean (MDRM), is an implementation of the minimum distance to mean (MDM) classification algorithm using Riemannian distance and Riemannian mean. This simple method shows comparable results with the reference method. The second method, named tangent space LDA (TSLDA), maps the covariance matrices onto the Riemannian tangent space where matrices can be vectorized and treated as Euclidean objects. Then, a variable selection procedure is applied in order to decrease dimensionality and a classification by LDA is performed. This latter method outperforms the reference method increasing the mean classification accuracy from 65.1% to 70.2%.
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Common Spatial Pattern revisited by Riemannian Geometry
2010Co-Authors: Alexandre Barachant, Marco Congedo, Stephane Bonnet, Christian JuttenAbstract:This paper presents a link between the well known Common Spatial Pattern (CSP) algorithm and Riemannian Geometry in the context of Brain Computer Interface (BCI). It will be shown that CSP spatial filtering and Log variance features extraction can be resumed as a computation of a Riemann distance in the space of covariances matrices. This fact yields to highlight several approximations with respect to the space topology. According to these conclusions, we propose an improvement of classical CSP method.
Yannick Berthoumieu - One of the best experts on this subject based on the ideXlab platform.
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transfer learning a Riemannian Geometry framework with applications to brain computer interfaces
IEEE Transactions on Biomedical Engineering, 2018Co-Authors: Paolo Zanini, Marco Congedo, Christian Jutten, Salem Said, Yannick BerthoumieuAbstract:Objective: This paper tackles the problem of transfer learning in the context of electroencephalogram (EEG)-based brain–computer interface (BCI) classification. In particular, the problems of cross-session and cross-subject classification are considered. These problems concern the ability to use data from previous sessions or from a database of past users to calibrate and initialize the classifier, allowing a calibration-less BCI mode of operation. Methods: Data are represented using spatial covariance matrices of the EEG signals, exploiting the recent successful techniques based on the Riemannian Geometry of the manifold of symmetric positive definite (SPD) matrices. Cross-session and cross-subject classification can be difficult, due to the many changes intervening between sessions and between subjects, including physiological, environmental, as well as instrumental changes. Here, we propose to affine transform the covariance matrices of every session/subject in order to center them with respect to a reference covariance matrix, making data from different sessions/subjects comparable. Then, classification is performed both using a standard minimum distance to mean classifier, and through a probabilistic classifier recently developed in the literature, based on a density function (mixture of Riemannian Gaussian distributions) defined on the SPD manifold. Results: The improvements in terms of classification performances achieved by introducing the affine transformation are documented with the analysis of two BCI datasets. Conclusion and significance: Hence, we make, through the affine transformation proposed, data from different sessions and subject comparable, providing a significant improvement in the BCI transfer learning problem.
Fabio Botelho - One of the best experts on this subject based on the ideXlab platform.
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a variational formulation for relativistic mechanics based on Riemannian Geometry and its application to the quantum mechanics context
arXiv: Analysis of PDEs, 2018Co-Authors: Fabio BotelhoAbstract:This article develops a variational formulation of relativistic nature applicable to the quantum mechanics context. The main results are obtained through basic concepts on Riemannian Geometry. Standards definitions such as vector fields and connection have a fundamental role in the main action establishment. In the last section, as a result of an approximation for the main formulation, we obtain the relativistic Klein-Gordon equation.
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a variational formulation for relativistic mechanics based on Riemannian Geometry and its application to the quantum mechanics context
Ciência e Natura, 2018Co-Authors: Fabio BotelhoAbstract:This article develops a variational formulation for relativistic quantum mechanics. The main results are obtained through a connection between relativistic and quantum mechanics. Such a connection is established through basic concepts on Riemannian Geometry and related extensions for the relativistic context.
