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 6.7 Ensembles of coupled oscillators in ML: Beyond Kuramoto models For the sake of completeness of the exposition, we mention some previous ML experiments with ensembles of coupled phase oscillators. Underline that most of these studies experimented with dynamical systems which do not belong to Kuramoto models. 6.7.1 Ensembles of coupled phase oscillators as associative memories 6.7.1 Ensembles of coupled phase oscillators as associative memories Several studies proposed to regard ensembles of coupled oscillators as associative memories [134, 135, 136]. This proposal was inspired by the Hopfield model, since ensembles of phase oscillators can be treated as generalizations of the famous Ising model. Ensembles of phase oscillators with symmetric couplings yield gradient flows on tori. Models proposed as associative memories are not Kuramoto models, since they include couplings through higherorder harmonics. Hence, they are not systems of geometric matrix Riccati ODE’s. are not One could think about designing associative memories based on extensions of ensembles of coupled oscillators to higher-dimensional manifolds. This might sound as a promising idea, but the corresponding theory is non-existent at the moment. Indeed, couplings through higher harmonics have been studied only for the ensembles of the classical phase oscillators, but not for generalized oscillators on spheres and other manifolds. Therefore, further investigations in this direction does not seem very promising, due to the total lack of theoretical background. 6.7.2 Swarms with plastic synapses for ML 6.7.2 Swarms with plastic synapses for ML An intriguing field of study in ML are NN architectures with plastic synapses (adaptive weights). The whole idea is, to a great extent, inspired by Neuroscience. We refer to the recent papers [137, 138] for fairly comprehensive reviews of NN’s with plastic synapses. Networks of this kind can be trained through backpropagation [137]. The central paradigm in networks with plastic synapses is the Hebbian learning rule. This rule, often summarized as "cells that fire together, wire together", imposes that synapses between two neurons get stronger, as their states get closer. Adaptive synapses obeying Hebbian rules are easily and naturally implemented in networks of Kuramoto oscillators [139]. Furthermore, it is straightforward to extend this model by introducing Kuramoto models on spheres with adaptive couplings [140]. Many models of this kind exhibit potential dynamics. One can enrich Kuramoto models on orthogonal groups and spheres by assuming that the couplings are adaptive and setting up the learning rule. In order to increase the representative power one can combine Hebbian and anti-Hebbian learning rules. In such a way one can construct the models representing coupled actions of the group SO(d) or complicated probability distributions on spheres. Notice that systems with adaptive couplings are not suitable for encoding hyperbolic isometries. Hence, actions of the Lorentz groups can not be modeled in this way. 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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Beyond Kuramoto Models: Associative Memory and Plastic Synapses in ML Ensembles

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