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Transport on Riemannian Manifold for Functional Connectivity-Based Classification [chapter]

Bernard Ng, Martin Dressler, Gaël Varoquaux, Jean Baptiste Poline, Michael Greicius, Bertrand Thirion
2014 Lecture Notes in Computer Science  
We present a Riemannian approach for classifying fMRI connectivity patterns before and after intervention in longitudinal studies.  ...  We show on real data that our approach provides significantly higher classification accuracy than directly using Pearson's correlation.  ...  Conclusions We presented a Riemannian approach for classifying functional connectivity patterns before and after interventions.  ... 
doi:10.1007/978-3-319-10470-6_51 fatcat:oqahza3l2bdc7bbc5nmwhocpse

Kernel-based classification for brain connectivity graphs on the Riemannian manifold of positive definite matrices

Luca Dodero, Ha Quang Minh, Marco San Biagio, Vittorio Murino, Diego Sona
2015 2015 IEEE 12th International Symposium on Biomedical Imaging (ISBI)  
In this paper, we propose a mathematical framework based on Riemannian geometry and kernel methods that can be applied to connectivity matrices for the classification task.  ...  An important task in connectomics studies is the classification of connectivity graphs coming from healthy and pathological subjects.  ...  In [3] , the concept of transport on the manifold of covariance matrices was successfully applied in longitudinal studies to classify changes in functional connectivity after a particular task .  ... 
doi:10.1109/isbi.2015.7163812 dblp:conf/isbi/DoderoMSMS15 fatcat:4eph2lewkbbg3a5ate7ujjhupa

Riemannian Manifold Optimization for Discriminant Subspace Learning [article]

Wanguang Yin, Zhengming Ma, Quanying Liu
2021 arXiv   pre-print
To address such a problem, in this paper, we propose a novel algorithm namely Riemannian-based discriminant analysis (RDA) for subspace learning.  ...  In order to obtain an explicit solution, we transform the traditional Euclidean-based methods to the Riemannian manifold space and use the trust-region method to learn the discriminant projection subspace  ...  Acknowledgements The authors would like to thank anonymous reviewers for their detailed and helpful comments. This work was funded in part by the National Natural Science  ... 
arXiv:2101.08032v3 fatcat:fahxx67cfrgolbyudqvbcz6fka

All regular Landsberg metrics are Berwald

Zoltán Imre Szabó
2008 Annals of Global Analysis and Geometry  
The proof of this statement is based on the following very simple idea: For a given Landsberg metric one constructs, first, a Riemannian metric tensor g i j ( p) by integrating the Landsberg metric tensor  ...  Or else, there is a much wider class of Finsler manifolds which allow such canonical volume forms on the base manifold.  ... 
doi:10.1007/s10455-008-9115-y fatcat:ub2f2ksd4bhqdaycsvur3wvbp4

Transport on Riemannian Manifold for Connectivity-Based Brain Decoding

Bernard Ng, Gael Varoquaux, Jean Baptiste Poline, Michael Greicius, Bertrand Thirion
2016 IEEE Transactions on Medical Imaging  
In this paper, we present a Riemannian approach for connectivity-based brain decoding.  ...  For this, we propose a matrix whitening transport, and compare it against parallel transport implemented via the Schild's ladder algorithm.  ...  In particular, many Transport on Riemannian Manifold for Connectivity-based Brain Decoding Bernard Ng, Gael Varoquaux, Jean Baptiste Poline, Michael Greicius, and Bertrand Thirion classifier learning  ... 
doi:10.1109/tmi.2015.2463723 pmid:26259016 fatcat:aebfjgwqqffrxkyas3cbmcvhl4

Geoopt: Riemannian Optimization in PyTorch [article]

Max Kochurov, Rasul Karimov, Serge Kozlukov
2020 arXiv   pre-print
Geoopt is a research-oriented modular open-source package for Riemannian Optimization in PyTorch.  ...  The core of Geoopt is a standard Manifold interface that allows for the generic implementation of optimization algorithms.  ...  To extend Geoopt, one should implement basic methods such as retraction or exponential map on the manifold, parallel or vector transport for tangent vectors, and make them properly broadcastable.  ... 
arXiv:2005.02819v5 fatcat:5duo56qmnfbffogiakzsr3jb3a

Semi-Riemannian Graph Convolutional Networks [article]

Bo Xiong, Shichao Zhu, Nico Potyka, Shirui Pan, Chuan Zhou, Steffen Staab
2021 arXiv   pre-print
We develop new geodesic tools that allow for extending neural network operations into geodesically disconnected semi-Riemannian manifolds.  ...  Non-Euclidean Riemannian manifolds provide specific inductive biases for embedding hierarchical or spherical data, but cannot align well with data of mixed topologies.  ...  Acknowledgments The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Bo Xiong.  ... 
arXiv:2106.03134v2 fatcat:2fk5vzyys5f2nkojrxdw5adtkq

Riemannian Holonomy Groups of Statistical Manifolds [article]

Didong Li, Huafei Sun, Chen Tao, Lin Jiu
2014 arXiv   pre-print
Normal distribution manifolds play essential roles in the theory of information geometry, so do holonomy groups in classification of Riemannian manifolds.  ...  , for all d∈N.  ...  Berger) The complete classification of possible holonomy groups for simply connected Riemannian manifolds which are irreducible and nonsymmetric is inTable 2(in Appendix).  ... 
arXiv:1401.5706v2 fatcat:nit2x4wkf5fczgn2nmvdbtqslq

Classification approach based on the product of riemannian manifolds from Gaussian parametrization space

Yannick Berthoumieu, Lionel Bombrun, Christian Germain, Salem Said
2017 2017 IEEE International Conference on Image Processing (ICIP)  
Classification approach based on the product of riemannian manifolds from Gaussian parametrization space.  ...  ABSTRACT This paper presents a novel framework for visual content classification using jointly local mean vectors and covariance matrices of pixel level input features.  ...  Random Variables on the Riemannian Manifold H m Let H m be a complete and simply connected m-dimensional Riemannian manifold. Let Y ∈ H m be a random variable with probability density function p(y).  ... 
doi:10.1109/icip.2017.8296272 dblp:conf/icip/BerthoumieuBGS17 fatcat:xuyx4mjpbbgf7hox53fx7pggn4

Deep Optimal Transport on SPD Manifolds for Domain Adaptation [article]

Ce Ju, Cuntai Guan
2022 arXiv   pre-print
The specific architecture in DOT enables it to learn an approximate optimal transport (OT) solution to the DA problems on SPD manifolds.  ...  The domain adaption (DA) problem on symmetric positive definite (SPD) manifolds has raised interest in the machine learning community because of the growing potential for the SPD-matrix representations  ...  Optimal Transport on Riemannian Manifolds Let (M, g) be a connected, compact, and C 3 smooth Riemannian manifold without boundary, equipped with a Riemannian distance d(x, y).  ... 
arXiv:2201.05745v1 fatcat:xz6xstvkabaofeq4gvqooojmvu

Metric Tensor and Christoffel Symbols Based 3D Object Categorization [chapter]

Syed Altaf Ganihar, Shreyas Joshi, Shankar Setty, Uma Mudenagudi
2015 Lecture Notes in Computer Science  
The metric tensor represents a geometrical signature of the 3D object in a Riemannian manifold.  ...  We model 3D object as a piecewise smooth Riemannian manifold and propose metric tensor and Christoffel symbols as a novel set of features.  ...  THEOREM 1 A Riemannian manifold (M, g) admits precisely one symmetric connection compatible with the metric.  ... 
doi:10.1007/978-3-319-16634-6_11 fatcat:33w4o3cz3ngq3oej2hd6aifdz4

Metric tensor and Christoffel symbols based 3D object categorization

Syed Altaf Ganihar, Shreyas Joshi, Shankar Shetty, Uma Mudenagudi
2014 ACM SIGGRAPH 2014 Posters on - SIGGRAPH '14  
The metric tensor represents a geometrical signature of the 3D object in a Riemannian manifold.  ...  We model 3D object as a piecewise smooth Riemannian manifold and propose metric tensor and Christoffel symbols as a novel set of features.  ...  THEOREM 1 A Riemannian manifold (M, g) admits precisely one symmetric connection compatible with the metric.  ... 
doi:10.1145/2614217.2630582 dblp:conf/siggraph/GaniharJSM14 fatcat:gylpehj77bbhpj54x4bzylfn4m

Some problems in differential geometry and topology

S K Donaldson
2008 Nonlinearity  
Again, one should qualify the degree to which this gives a precise classification.  ...  Low-dimensional topology and symplectic topology The general setting here is that one would like to understand the classification of manifolds, and related objects such as knots in 3-space.  ...  Special holonomy and calibrated geometry The Levi-Civita connection of a Riemannian manifold M defines the operation of parallel transport of tangent vectors along paths.  ... 
doi:10.1088/0951-7715/21/9/t02 fatcat:bpwaapensjenvoh57bb2sf45ni

Semi-Riemannian cones [article]

Thomas Leistner
2020 arXiv   pre-print
We will use these results to give a new proof of the classification results for Riemannian manifolds with imaginary Killing spinors and Lorentzian manifolds with real Killing spinors which are due to Baum  ...  This is false in general for indefinite cones but the structures induced on the cone by holonomy invariant subspaces can be used to study the geometry on the base of the cone.  ...  One reason for considering semi-Riemannian cones is that some systems of PDE on the base correspond to PDE on the cone where they sometimes are easier to study.  ... 
arXiv:2001.07349v1 fatcat:blodoijcgjd3fmfrtjrpzih7ym

Lévy Laplacians, holonomy group and instantons on 4-manifolds [article]

Boris O. Volkov
2021 arXiv   pre-print
The connection between Yang–Mills gauge fields on 4-dimensional orientable compact Riemannian manifolds and modified Lévy Laplacians is studied.  ...  There is a modified Lévy Laplacian such that a parallel transport in some vector bundle over the 4-manifold is a solution of the Laplace equation for this modified Lévy Laplacian if and only if the connection  ...  Volovich for helpful discussions.  ... 
arXiv:2107.11215v1 fatcat:sd7hs53f55gv5n6v6cuomaqhkm
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