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
2.2 Probabilistic modeling and inference in DL
Learning in general can be viewed as a process of updating certain beliefs about the state of the world based on the new information. This abstract point of view underlies the broad field of Probabilistic ML [17]. In this Subsection we mention certain aspects of this field which are the most relevant for the present study.
The general idea of updating beliefs can be formalized as learning an optimal (according to a certain criterion) probability distribution. This further implies that implementation of probabilistic ML algorithms involves optimization over spaces of probability distributions. Therefore, gradient flows on spaces of probability measures [18, 19] are essential ingredients of probabilistic modeling in ML. The notion of gradient flow requires the metric structure. The distance between two probability measures should represent the degree of difficulty to distinguish between them provided that a limited number of samples is available. Metric on spaces of probability measures are induced by the Hessians of various divergence functions [20, 21]. The classical (and parametric invariant) choice is the Kullback-Leibler divergence (KL divergence), also referred to as relative entropy. This divergence induces the Fisher (or Fisher-Rao) information metric on spaces of probability measures thus turning them into statistical manifolds [20]. When optimizing over a family of probability distributions, Euclidean (or, so called, “vanilla”) gradient is inappropriate. Ignoring of this fact, leads to inaccurate or incorrect algorithms. Instead, one should use the gradient w.r. to Fisher information metric, which is named natural gradient [22, 23, 24]. In RL this must be taken into account when designing stochastic policies. In particular, well known actor-critic algorithm has been modified in order to respect geometry of the space of probability distributions [25].
Another way of introducing metric on spaces of probability distributions is the Wasserstein metric. Fokker-Planck equations are gradient flows in the Wasserstein metric. The potential function for these flows is the KL divergence between the instant and (unknown) stationary distribution [26]. Yet another metric sometimes used in ML is the Stein metric [27].
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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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