A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups (2104.09459v1)

Abstract: Symmetries and equivariance are fundamental to the generalization of neural networks on domains such as images, graphs, and point clouds. Existing work has primarily focused on a small number of groups, such as the translation, rotation, and permutation groups. In this work we provide a completely general algorithm for solving for the equivariant layers of matrix groups. In addition to recovering solutions from other works as special cases, we construct multilayer perceptrons equivariant to multiple groups that have never been tackled before, including $\mathrm{O}(1,3)$, $\mathrm{O}(5)$, $\mathrm{Sp}(n)$, and the Rubik's cube group. Our approach outperforms non-equivariant baselines, with applications to particle physics and dynamical systems. We release our software library to enable researchers to construct equivariant layers for arbitrary matrix groups.

• The paper introduces a practical method that reduces equivariance constraints to a finite set of linear equations (M + D constraints).
• It develops a polynomial-time algorithm and a general architecture with a bilinear layer applicable to diverse symmetry groups.
• Empirical results demonstrate superior data efficiency and accurate modeling of physical systems compared to non-equivariant baselines.

Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups

This paper presents a significant advancement in the development of neural network architectures by introducing a practical method for constructing equivariant multilayer perceptrons (EMLPs) applicable to arbitrary matrix groups. Equivariance in neural networks allows for transformations of the input space that preserve the output structure, which is crucial for tasks involving symmetries, such as image recognition and physical system modeling.

Theoretical Contributions

The authors establish a general algorithm for designing equivariant layers in multilayer perceptrons (MLPs) by utilizing matrix groups. The originality of this approach lies in its generality, accommodating any finite-dimensional representation of a symmetry group. This extends beyond the conventional focus on commonly studied groups like translations, rotations, and permutations.

The paper offers several key contributions to the theory and application of equivariant neural networks:

1. Equivariance Reduction: The authors prove that the constraints for equivariance in matrix groups can be distilled into a finite set of linear constraints, precisely $M + D$ constraints, where $M$ is the number of discrete generators and $D$ is the group's dimension. This reduction significantly simplifies the process of constructing equivariant layers.
2. Polynomial Time Algorithm: They present a polynomial time algorithm for determining these equivariant constraints in finite-dimensional representations, streamlining the computation needed for practical implementation.
3. General Equivariant Architecture: By incorporating a bilinear layer, the authors develop a generalized architecture for EMLP that is versatile across different groups by specifying the group generators. The EMLP framework is applicable to symmetries not previously addressed, such as the orthogonal group in five dimensions, the full Lorentz group, and even the Rubik’s cube group.

Empirical Results

EMLP demonstrates superior performance compared to non-equivariant baselines in several domains, particularly in tasks involving particle physics and dynamical systems. The paper details several experiments:

• Data Efficiency: EMLPs exhibit marked data efficiency, significantly outperforming standard MLPs, even when the latter are enhanced with data augmentation techniques to mimic equivariance. This is evident in tasks such as predicting moments of inertia and modeling Lorentz-invariant scattering.
• Dynamical Systems: In the context of learning dynamics, EMLPs capture symmetries inherent in physical systems more accurately. For instance, applying EMLP to a double spring pendulum system illustrates how computed symmetries like angular momentum conservation are respected.

Practical Implications and Future Directions

The method proposed in this paper has substantial implications for both theoretical advancements and practical applications:

• Broad Applicability: By supporting arbitrary matrix groups, the architecture can be adapted to exploit domain-specific symmetries across numerous applications, from physics to computer graphics.
• Accelerated Research Development: The provided software library allows researchers across fields to more readily implement and experiment with equivariant neural networks tailored to their specific problem domains.
• Better Modeling of Physical Systems: With built-in symmetries, models can avoid overfitting to spurious patterns, improving generalization, particularly in data-limited settings.

Future Directions: Extending EMLPs to even larger and more complex representations, and optimizing for computational efficiency, will broaden their applicability in deep learning scenarios requiring diverse input structures. Research can also explore the intersections of equivariance with unsupervised learning and reinforcement learning, domains where structured input may play a critical role. Enhancing parallelized implementations for specific equivariant architectures can facilitate their use in high-performance applications akin to traditional convolutional networks.

In conclusion, this work provides a foundational approach to integrating symmetries in neural networks, offering a powerful tool for enhancing the generalization and efficiency of AI across complex domains.