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 6.2 Swarms on manifolds and directional statistics in RL References on RL on non-Euclidean spaces are very sparse. Researchers only recently started to apply RL techniques to the learning over data sets with non-Euclidean geometries. Reward-based update of deterministic and stochastic policies on Riemannian manifolds has potential applications in different fields. Hence, Markov decision processes with non-Euclidean space of states (and actions) present an important challenge for ML. However, rather than developing the general theory, it seems more advantageous to start with some paradigmatic examples in order to elaborate conceptual approaches. Two recent studies [59, 64] considered RL algorithms on spheres and special orthogonal groups. Both studies proposed the Bingham distribution for implementation of stochastic policies. Parametrization of stochastic policies by spherical Cauchy and von Mises-Fisher distributions in RL is still to be explored. In our point of view, the idea of using spherical Cauchy distributions in RL has a great potential, given all nice properties of this family. We will discuss this idea on some illustrative examples in the next Section. We also mention a very recent paper [118] on hyperbolic RL. This paper proposes deep RL algorithms for designing more efficient policies that model latent representations in the hyperbolic space. Like most of ML algorithms in hyperbolic geometries this method exploits the framework of gyrovector spaces for operations in hyperbolic spaces. 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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Reinforcement Learning on Non-Euclidean Spaces: Swarms, Spheres, and Hyperbolic RL

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