Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Abstract and 1. Introduction Some recent trends in theoretical ML
2.1 Deep Learning via continuous-time controlled dynamical system
2.2 Probabilistic modeling and inference in DL
2.3 Deep Learning in non-Euclidean spaces
2.4 Physics Informed ML


Kuramoto model
3.1 Kuramoto models from the geometric point of view
3.2 Hyperbolic geometry of Kuramoto ensembles
3.3 Kuramoto models with several globally coupled sub-ensembles


Kuramoto models on higher-dimensional manifolds
4.1 Non-Abelian Kuramoto models on Lie groups
4.2 Kuramoto models on spheres
4.3 Kuramoto models on spheres with several globally coupled sub-ensembles
4.4 Kuramoto models as gradient flows
4.5 Consensus algorithms on other manifolds


Directional statistics and swarms on manifolds for probabilistic modeling and inference on Riemannian manifolds
5.1 Statistical models over circles and tori
5.2 Statistical models over spheres
5.3 Statistical models over hyperbolic spaces
5.4 Statistical models over orthogonal groups, Grassmannians, homogeneous spaces


Swarms on manifolds for DL
6.1 Training swarms on manifolds for supervised ML
6.2 Swarms on manifolds and directional statistics in RL
6.3 Swarms on manifolds and directional statistics for unsupervised ML
6.4 Statistical models for the latent space
6.5 Kuramoto models for learning (coupled) actions of Lie groups
6.6 Grassmannian shallow and deep learning
6.7 Ensembles of coupled oscillators in ML: Beyond Kuramoto models


Examples
7.1 Wahba’s problem
7.2 Linked robot’s arm (planar rotations)
7.3 Linked robot’s arm (spatial rotations)
7.4 Embedding multilayer complex networks (Learning coupled actions of Lorentz groups)


Conclusion and References Some recent trends in theoretical ML
2.1 Deep Learning via continuous-time controlled dynamical system
2.2 Probabilistic modeling and inference in DL
2.3 Deep Learning in non-Euclidean spaces
2.4 Physics Informed ML Some recent trends in theoretical ML Some recent trends in theoretical ML Some recent trends in theoretical ML 2.1 Deep Learning via continuous-time controlled dynamical system 2.1 Deep Learning via continuous-time controlled dynamical system 2.2 Probabilistic modeling and inference in DL 2.2 Probabilistic modeling and inference in DL 2.3 Deep Learning in non-Euclidean spaces 2.3 Deep Learning in non-Euclidean spaces 2.4 Physics Informed ML 2.4 Physics Informed ML Kuramoto model
3.1 Kuramoto models from the geometric point of view
3.2 Hyperbolic geometry of Kuramoto ensembles
3.3 Kuramoto models with several globally coupled sub-ensembles Kuramoto model Kuramoto model Kuramoto model 3.1 Kuramoto models from the geometric point of view 3.1 Kuramoto models from the geometric point of view 3.2 Hyperbolic geometry of Kuramoto ensembles 3.2 Hyperbolic geometry of Kuramoto ensembles 3.3 Kuramoto models with several globally coupled sub-ensembles 3.3 Kuramoto models with several globally coupled sub-ensembles Kuramoto models on higher-dimensional manifolds
4.1 Non-Abelian Kuramoto models on Lie groups
4.2 Kuramoto models on spheres
4.3 Kuramoto models on spheres with several globally coupled sub-ensembles
4.4 Kuramoto models as gradient flows
4.5 Consensus algorithms on other manifolds Kuramoto models on higher-dimensional manifolds Kuramoto models on higher-dimensional manifolds Kuramoto models on higher-dimensional manifolds 4.1 Non-Abelian Kuramoto models on Lie groups 4.1 Non-Abelian Kuramoto models on Lie groups 4.2 Kuramoto models on spheres 4.2 Kuramoto models on spheres 4.3 Kuramoto models on spheres with several globally coupled sub-ensembles 4.3 Kuramoto models on spheres with several globally coupled sub-ensembles 4.4 Kuramoto models as gradient flows 4.4 Kuramoto models as gradient flows 4.5 Consensus algorithms on other manifolds 4.5 Consensus algorithms on other manifolds Directional statistics and swarms on manifolds for probabilistic modeling and inference on Riemannian manifolds
5.1 Statistical models over circles and tori
5.2 Statistical models over spheres
5.3 Statistical models over hyperbolic spaces
5.4 Statistical models over orthogonal groups, Grassmannians, homogeneous spaces Directional statistics and swarms on manifolds for probabilistic modeling and inference on Riemannian manifolds Directional statistics and swarms on manifolds for probabilistic modeling and inference on Riemannian manifolds Directional statistics and swarms on manifolds for probabilistic modeling and inference on Riemannian manifolds 5.1 Statistical models over circles and tori 5.1 Statistical models over circles and tori 5.2 Statistical models over spheres 5.2 Statistical models over spheres 5.3 Statistical models over hyperbolic spaces 5.3 Statistical models over hyperbolic spaces 5.4 Statistical models over orthogonal groups, Grassmannians, homogeneous spaces 5.4 Statistical models over orthogonal groups, Grassmannians, homogeneous spaces Swarms on manifolds for DL
6.1 Training swarms on manifolds for supervised ML
6.2 Swarms on manifolds and directional statistics in RL
6.3 Swarms on manifolds and directional statistics for unsupervised ML
6.4 Statistical models for the latent space
6.5 Kuramoto models for learning (coupled) actions of Lie groups
6.6 Grassmannian shallow and deep learning
6.7 Ensembles of coupled oscillators in ML: Beyond Kuramoto models Swarms on manifolds for DL Swarms on manifolds for DL Swarms on manifolds for DL 6.1 Training swarms on manifolds for supervised ML 6.1 Training swarms on manifolds for supervised ML 6.2 Swarms on manifolds and directional statistics in RL 6.2 Swarms on manifolds and directional statistics in RL 6.3 Swarms on manifolds and directional statistics for unsupervised ML 6.3 Swarms on manifolds and directional statistics for unsupervised ML 6.4 Statistical models for the latent space 6.4 Statistical models for the latent space 6.5 Kuramoto models for learning (coupled) actions of Lie groups 6.5 Kuramoto models for learning (coupled) actions of Lie groups 6.6 Grassmannian shallow and deep learning 6.6 Grassmannian shallow and deep learning 6.7 Ensembles of coupled oscillators in ML: Beyond Kuramoto models 6.7 Ensembles of coupled oscillators in ML: Beyond Kuramoto models Examples
7.1 Wahba’s problem
7.2 Linked robot’s arm (planar rotations)
7.3 Linked robot’s arm (spatial rotations)
7.4 Embedding multilayer complex networks (Learning coupled actions of Lorentz groups) Examples Examples 7.1 Wahba’s problem 7.1 Wahba’s problem 7.2 Linked robot’s arm (planar rotations) 7.2 Linked robot’s arm (planar rotations) 7.3 Linked robot’s arm (spatial rotations) 7.3 Linked robot’s arm (spatial rotations) 7.4 Embedding multilayer complex networks (Learning coupled actions of Lorentz groups) 7.4 Embedding multilayer complex networks (Learning coupled actions of Lorentz groups) Conclusion and References Conclusion and References Conclusion and References Conclusion and References 6 Swarms on manifolds for DL We have presented a broad class of models that generate trajectories on various manifolds. Given a particular setup, an appropriate model can be chosen and trained in order to learn an optimal (in a certain sense) trajectory on a specific manifold. Training these dynamical systems boils down to the estimation of parameters: coupling strengths, phase shifts, initial positions of (generalized) oscillators, etc. Noiseless models with global coupling generate trajectories on invariant statistical manifolds (wrapped Cauchy, spherical Cauchy, etc.). On the other hand, some important distributions (von Mises, hyperbolic von Mises, von Mises-Fisher) arise as stationary distributions in Kuramoto models with noise. In this Section we briefly discuss ideas on implementation of the corresponding algorithms. Author:
(1) Vladimir Jacimovic, Faculty of Natural Sciences and Mathematics, University of Montenegro Cetinjski put bb., 81000 Podgorica Montenegro (vladimirj@ucg.ac.me). Author: Author: (1) Vladimir Jacimovic, Faculty of Natural Sciences and Mathematics, University of Montenegro Cetinjski put bb., 81000 Podgorica Montenegro (vladimirj@ucg.ac.me). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

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Swarms on Manifolds for Deep Learning: Training Kuramoto Models and Trajectory Learning

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