Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks (2409.11772v2)

Abstract: There has been much recent interest in designing neural networks (NNs) with relaxed equivariance, which interpolate between exact equivariance and full flexibility for consistent performance gains. In a separate line of work, structured parameter matrices with low displacement rank (LDR) -- which permit fast function and gradient evaluation -- have been used to create compact NNs, though primarily benefiting classical convolutional neural networks (CNNs). In this work, we propose a framework based on symmetry-based structured matrices to build approximately equivariant NNs with fewer parameters. Our approach unifies the aforementioned areas using Group Matrices (GMs), a forgotten precursor to the modern notion of regular representations of finite groups. GMs allow the design of structured matrices similar to LDR matrices, which can generalize all the elementary operations of a CNN from cyclic groups to arbitrary finite groups. We show GMs can also generalize classical LDR theory to general discrete groups, enabling a natural formalism for approximate equivariance. We test GM-based architectures on various tasks with relaxed symmetry and find that our framework performs competitively with approximately equivariant NNs and other structured matrix-based methods, often with one to two orders of magnitude fewer parameters.

• The paper introduces group matrices as a novel framework that extends classical convolutions to arbitrary discrete groups for enhanced network symmetry.
• It generalizes CNN operations by adapting convolution, striding, and pooling techniques to work over structured matrices with low displacement rank metrics.
• Experimental results validate the approach, showing competitive performance and reduced parameter counts in dynamics prediction and image classification tasks.

Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks

"Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks," authored by Ashwin Samudre, Mircea Petrache, Brian D. Nord, and Shubhendu Trivedi, presents a comprehensive paper on the integration of symmetry properties into neural networks through a novel framework that leverages the concept of group matrices (GMs). This paper addresses the intersection of equivariant learning and efficient neural network design by introducing symmetry-based structured matrices to create lightweight models with competitive performance and approximately equivariant properties.

Overview

The paper begins by situating its contributions within two established research areas: symmetry-aware neural networks (NNs) and structured parameter matrices. Symmetry-aware NNs, which incorporate group equivariance, have shown substantial benefits in various domains. However, the rigid equivariance constraints can sometimes degrade model performance due to the mismatch between data symmetries and model symmetries. On the other hand, structured parameter matrices, particularly those with low displacement rank (LDR), have enabled the development of resource-efficient NNs by approximating dense weight matrices with structured alternatives like Toeplitz or circulant matrices.

Main Contributions

The key contributions of the paper can be delineated as follows:

1. Introduction of Group Matrices (GMs): The authors propose the use of group matrices as a foundational tool to extend classical convolutions to arbitrary finite groups. This extension not only covers cyclic groups but generalizes to any discrete groups, facilitating the construction of equivariant NNs. GMs are constructed such that they resemble LDR matrices, thus combining the benefits of structured matrices with symmetry-aware designs.
2. Generalization of CNN Operations: The framework encompasses all elementary operations of classical CNNs, including convolution, striding, and pooling, and adapts them to operate over arbitrary discrete groups using GMs. The paper details efficient procedures for constructing these generalized operations. It also describes how pooling operations can be derived using the notion of group coarsening appropriate for discrete groups.
3. Approximate Equivariance: The novel framework introduces flexibility by allowing approximate rather than perfect equivariance through the incorporation of low displacement rank metrics. This principled relaxation allows models to adapt to data symmetries more effectively while maintaining computational efficiencies. The authors quantify the error due to approximate equivariance and extend classical LDR theory to accommodate general discrete groups.
4. Experimental Validation: The paper provides comprehensive experimental results demonstrating the proposed framework's effectiveness across various tasks, highlighting the substantial reduction in parameter counts while maintaining competitive performance. Metrics such as mean squared error (MSE) on dynamics prediction tasks and classification accuracy on image datasets validate the efficacy of GM-CNNs compared to baseline approx-equivariant methods and structured matrix-based frameworks.

Implications and Future Work

Both practical and theoretical implications emerge from this work. Practically, the framework offers a pathway to designing NNs that are lightweight and adaptable, making them suitable for deployment in resource-constrained environments while still maintaining robust performance under symmetry considerations. Theoretically, the integration of group matrices expands the understanding of symmetry applications in deep learning, providing a foundation for further explorations into other types of symmetries and extensions to homogeneous spaces and infinite groups.

Future developments could involve extending this approach to continuous groups, enhancing the framework's capacity for a broader range of applications. Additionally, further investigating tensorization operations, as mentioned in the appendix, could improve scalability and comprehensive applicability in larger datasets and more complex model architectures.

Conclusion

This paper advances the field of symmetry-aware NNs by merging them with structured matrix representations to produce efficient, approximately equivariant networks. The introduction of group matrices and their comprehensive integration into network operations stands as a significant contribution, enabling the modeling of symmetries in a resource-efficient manner. These innovations hold promise for both theoretical advancements and practical applications, ensuring that the models both respect symmetries present in the data and operate efficiently across various domains.