Machine Learning In Non-Euclidean Space Enabled By The Kuramoto Model

Computer Vision 04/12/2024

3 main points
✔️ Proposes a method for machine learning on non-Euclidean data sets using the Kramoto model and its extension to higher dimensions.
✔️ Argues for the use of probability distribution families invariant to the action of certain symmetric groups as probability models in geometric deep learning.
✔️ The Cramotto oscillator and swarm models are used to achieve learning on the coordinated action of transformation groups and on data on spherical and hyperbolic multiple disks.

Kuramoto Oscillators and Swarms on Manifolds for Geometry Informed Machine Learning
written by Vladimir Jacimovic
(Submitted on 15 May 2024)
Comments: Published on arxiv.
Subjects: Machine Learning (cs.LG); Mathematical Physics (math-ph); Adaptation and Self-Organizing Systems (nlin.AO)

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The images used in this article are from the paper, the introductory slides, or were created based on them.

Summary

Machine learning is the science of finding patterns in data, but many previous models have assumed Euclidean, or flat, spaces. However, recent research has shown that many data sets have properties that are inherently non-Euclidean, or curved. For example, spherical and hyperbolic spaces. To address this, a new approach has been proposed using the Kuramoto model. The Kuramoto model describes the collective motion of oscillators, and by extending it, data can be learned on non-Euclidean spaces.

Furthermore, these models allow for efficient machine learning using probability distributions with specific symmetries. This allows for more flexible and powerful data analysis that is not limited by the conventional Euclidean space. This paper details the theoretical background and practical applications of these new approaches.

Related Research

There are several interesting research directions in the theoretical development of machine learning. Some recent noteworthy topics are listed below.

Deep Learning with Continuous-Time Controlled Dynamical Systems

The approach proposed by Weinan E in 2017 introduced a new way to realize neural networks (NNs) as continuous-time controlled dynamical systems. The idea is based on the observation that conventional NNs can be interpreted as an Eulerian discretization of a controlled ordinary differential equation (ODE). This allows the weights to be replaced by control functions and training using Pontryagin's maximum principle. This research has produced an outcome called Neural ODE, which aims to formalize many machine learning tasks as optimal control problems.

Probability Modeling and Inference

In machine learning, learning is generally viewed as the process of updating beliefs based on new information. The goal is thereby to learn an optimal probability distribution. The gradient flow over the space of probability distributions forms an essential part of this process. For example, the use of natural gradients based on the Fisher information metric improves the efficiency and accuracy of probabilistic modeling.

Deep Learning in Non-Euclidean Space

Many datasets have non-Euclidean geometry, which tends to lead to inaccurate algorithms if ignored. For example, when learning to rotate in 3D space, traditional NNs that assume Euclidean space are not applicable. Instead, geometric methods for handling data in curved spaces, such as Riemannian manifolds and hyperbolic geometry, are required.

Physics-Based Machine Learning

It is an approach to designing machine learning algorithms using physical laws, such as conservation laws and symmetries. Models based on physical laws provide efficient, transparent, and robust algorithms. The concepts of energy and entropy have also played a central role in early machine learning algorithms.

Each of these research directions expands the possibilities of machine learning in its own way. In particular, methods based on non-Euclidean geometry and approaches that use physical laws will become increasingly important.

Proposed Method

Here we propose a new method for machine learning on non-Euclidean spaces using the Kramoto model and its generalization to higher dimensions. Specifically, we take the following approach.

Extension of the Clamotte model

The Kuramoto model was introduced in 1975 to describe oscillator synchronization phenomena. This model is extended to learn data on higher dimensional manifolds such as spheres and Lie groups. For example, it can learn the action of a particular symmetry group and handle data on spherical or hyperbolic multiple disks.

Use of Stochastic Models

As a probabilistic model in geometric deep learning, we use probability distribution families that are invariant to the action of a particular symmetry group. This allows for efficient probabilistic modeling and inference on non-Euclidean data sets. For example, when learning distributions on hyperbolic spaces or spheres, these probability distributions can be used for more accurate modeling.

Learning Herd Models

We use Cramotto oscillators and group models to learn the coordinated action of transformation groups. This allows us to learn the combined action of certain transformation groups (e.g., special orthogonal groups, unitary groups, Lorenz groups, etc.). These models can also be used to effectively handle data in spherical and hyperbolic spaces.

Relationship between Noise and Distribution

By adding noise to the Clamot model, specific probability distributions (e.g., von Mises or wrapped Cauchy distributions) can be learned. This allows learning under the influence of noise, starting from an initial uniform distribution. This approach allows for efficient estimation of complex distributions.

Experiment

In this study, a series of experiments were conducted to demonstrate data learning on a non-Euclidean space using the proposed Kuramoto model and its higher dimensional generalization. The following results were obtained as a result of the experiments.

Learning Data on A Sphere

The extension of the Kuramoto model on the sphere was confirmed to be efficient in learning datasets on the sphere. In particular, the accuracy was improved for classification problems on spheres compared to conventional methods based on Euclidean space.

Clustering in Hyperbolic Space

In clustering experiments in hyperbolic space, the Kuramoto model was able to capture the latent hierarchical structure of the data, resulting in highly accurate clustering. This result is particularly useful for hierarchical data sets, such as natural language processing and molecular structure analysis.

Learning Coordinated Action of Transformation Groups

Experiments on learning the action of transformation groups such as special orthogonal groups and unitary groups showed that the proposed model can learn these complex transformations with high accuracy. The results are expected to be applied in fields where rotations and transformations play an important role, such as robotics and computer vision.

Consideration

Advantages of Learning in Non-Euclidean Spaces

Using the Kuramoto model and its higher dimensional generalization, it is clear that the proposed method can effectively learn properties of non-Euclidean spaces that are not captured by conventional methods based on Euclidean spaces. In particular, the proposed method was shown to be useful for data analysis in curved spaces such as spherical and hyperbolic spaces.

Importance of Stochastic Models

The proposed method allows efficient stochastic modeling and inference on non-Euclidean spaces by utilizing probability distributions that are invariant to the action of a particular symmetry group. This has been confirmed to achieve higher accuracy and efficiency than conventional methods.

Conclusion

In this study, we proposed a new method for learning data in non-Euclidean space using the Kuramoto model and its generalization to higher dimensions, and demonstrated its effectiveness. This enables us to effectively handle complex data structures that cannot be handled by conventional methods based on Euclidean space.

In the future, along with strengthening the theoretical foundation, they hope to apply the model to real-world problems such as robotics, natural language processing, and molecular structure analysis. They also plan to extend the model to other non-Euclidean spaces and improve computational efficiency. They hope that these results will lead to further technological innovation and diverse applications.

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