Table of Links
Abstract and 1. Introduction
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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
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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
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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
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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
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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
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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)
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Conclusion and References
2.1 Deep Learning via continuous-time controlled dynamical system
In 2017. Weinan E proposed new architectures of NN’s realized through the continuous-time controlled dynamical systems [10]. This proposal was motivated by the previous observations that NN’s (most notably, ResNets) can be regarded as Euler discretizations of controlled ODE’s. In parallel, a number of studies [11, 12, 13] enhanced and expanded theoretical foundations of ML by adapting classical control-theoretic techniques to the new promising field of applications.
This line of research resulted in a tangible outcome which was named Neural ODE [14]. The underlying idea is to formalize some ML tasks as optimal control problems. In fact, deep limits of ResNets with constant weights yield continuous-time dynamical systems [15]. In such a setup weights of the NN are replaced by control functions. Training of the model is realized through minimization of the total error (or total loss) using the Pontryagin’s maximum principle. Backpropagation corresponds to the adjoint ODE which is solved backwards in time.
A similar way of encoding maps underlies the concept of continuous-time normalizing flows [16]. Normalizing flows are dynamical systems, usually described by ODE’s or PDE’s. These systems are trained with the goal of learning a sequence (or a flow) of invertible maps between the observed data originating from an unknown complicated target probability distribution and some simple (typically Gaussian) distribution. Once the normalizing flow is trained, the target distribution is approximated. The model is capable of generalizing the observed data and making predictions by sampling from the simple distribution and mapping the samples along the learned flow.
We have mentioned two concepts (neural ODE and normalizing flows) that recently had a significant impact. Their success reflects the general trend of growing interest in control-theoretic point of view on ML. Most of theoretical advances in Reinforcement Learning (RL) rely on Control Theory (CT) [12, 13]. As theoretical foundations of RL are being established, the boundary between RL and CT is getting blurred.
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Author:
(1) Vladimir Jacimovic, Faculty of Natural Sciences and Mathematics, University of Montenegro Cetinjski put bb., 81000 Podgorica Montenegro ([email protected]).
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This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license.
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