Discovering Nonlinear Symmetries in High-Dimensional Data Using Latent Space Representations

Core Concepts

LaLiGAN, a novel generative modeling framework, can discover nonlinear symmetries in high-dimensional data by decomposing the group action into nonlinear mappings between the data space and a latent space, and a linear group representation in the latent space.

Abstract

The paper proposes a novel generative modeling framework called LaLiGAN for discovering nonlinear symmetries in high-dimensional data. The key insight is that nonlinear group transformations can be decomposed into nonlinear mappings between the data space and a latent space, and a linear group representation in the latent space. LaLiGAN learns this decomposition by jointly optimizing the nonlinear mappings (encoder and decoder) and the linear group representation in the latent space. The authors provide theoretical guarantees that this decomposition can approximate any nonlinear symmetry under certain conditions. The discovered latent space symmetries can be used for various downstream tasks, such as: Equation discovery: The latent space learned by LaLiGAN leads to simpler governing equations and improved long-term prediction accuracy compared to using an autoencoder alone. Learning equivariant representations: When the symmetry group is known, LaLiGAN can learn the corresponding group equivariant representation without any knowledge of the group element associated with each data point. The authors demonstrate the effectiveness of LaLiGAN on several dynamical systems with complicated nonlinear symmetries, including reaction-diffusion, nonlinear pendulum, and Lotka-Volterra systems. The discovered latent space symmetries accurately capture the intrinsic structure of these high-dimensional systems.

Stats

The paper does not provide any specific numerical data or statistics. The key results are presented through visualizations of the latent space representations and the discovered governing equations.

Quotes

"Equivariant neural networks require explicit knowledge of the symmetry group. Automatic symmetry discovery methods aim to relax this constraint and learn invariance and equivariance from data." "Our key insight is that nonlinear group transformations can be decomposed into nonlinear mappings between data space and latent space, and a linear group representation in the latent space." "The significance of latent space symmetry discovery is multifold. From the perspective of symmetry discovery, it further expands the search space of symmetries beyond linear group actions."

Key Insights Distilled From

Latent Space Symmetry Discovery

by Jianke Yang,... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2310.00105.pdf

Deeper Inquiries

How can the theoretical guarantees provided in Theorem 4.1 be extended to handle more general group actions, such as non-compact groups or infinite-dimensional Lie groups

The theoretical guarantees provided in Theorem 4.1 can be extended to handle more general group actions by considering a broader class of symmetry groups. For non-compact groups or infinite-dimensional Lie groups, the key lies in adapting the decomposition of the group action to accommodate the specific properties of these groups. For non-compact groups, the decomposition of the group action may involve continuous transformations that do not have a finite-dimensional representation. In this case, the neural networks parametrizing the nonlinear mappings between the data space and the latent space would need to be able to capture the continuous nature of the group action. The Lie algebra basis and Lie group representations would need to be generalized to handle the infinite-dimensional structure of the group. For infinite-dimensional Lie groups, such as those associated with infinite-dimensional manifolds or function spaces, the decomposition of the group action would need to account for the infinite-dimensional nature of the group transformations. The neural networks involved in the decomposition would need to be capable of handling functions or distributions as inputs and outputs, rather than finite-dimensional vectors. In summary, extending the theoretical guarantees to handle more general group actions involves adapting the decomposition framework to accommodate the specific properties of non-compact or infinite-dimensional Lie groups, ensuring that the neural networks can effectively capture the continuous or infinite-dimensional nature of the group transformations.

What other physical properties, beyond symmetries, can be incorporated into the latent space representation learned by LaLiGAN, and how would that affect the downstream tasks

Beyond symmetries, LaLiGAN can incorporate other physical properties into the latent space representation to enhance downstream tasks. Some of these properties include conservation laws, energy constraints, or structural constraints specific to the domain of interest. By incorporating these additional physical properties into the latent space representation, LaLiGAN can provide a more structured and meaningful representation that aligns with the underlying physics of the system. Incorporating conservation laws, for example, can ensure that the learned latent space representations respect fundamental principles of energy conservation or momentum conservation. This can lead to more physically meaningful representations and improve the accuracy of downstream tasks such as equation discovery or long-term forecasting. Similarly, incorporating energy constraints can help regularize the latent space representations to ensure that the learned symmetries and dynamics align with the energy landscape of the system. This can lead to more stable and physically realistic representations that capture the essential dynamics of the system. By integrating these additional physical properties into the latent space representation, LaLiGAN can provide a more comprehensive and interpretable representation that captures not only symmetries but also other fundamental aspects of the underlying physical system.

Can the latent space symmetry discovery framework be applied to other domains beyond dynamical systems, such as computer vision or natural language processing, and what are the potential challenges in those applications

The latent space symmetry discovery framework can be applied to other domains beyond dynamical systems, such as computer vision or natural language processing, with some adaptations and challenges. In computer vision, the framework can be used to discover latent symmetries in image data, such as rotational symmetries, scale invariance, or translation invariance. By learning structured latent representations that capture these symmetries, the framework can enhance tasks like image classification, object detection, or image generation. However, challenges may arise in handling the high-dimensional and complex nature of image data, as well as ensuring that the learned symmetries align with the visual characteristics of the data. In natural language processing, the framework can be applied to discover latent symmetries in text data, such as semantic relationships, syntactic structures, or linguistic patterns. By learning latent representations that encode these symmetries, the framework can improve tasks like language modeling, sentiment analysis, or machine translation. Challenges in this domain may include dealing with the sequential and hierarchical nature of text data, as well as ensuring that the learned symmetries capture the nuances of language semantics and syntax. Overall, while the latent space symmetry discovery framework can be extended to other domains, the challenges lie in adapting the framework to the specific characteristics and complexities of the data in those domains, as well as ensuring that the learned symmetries are meaningful and beneficial for the downstream tasks.