Learning Equivariant Functions with Probabilistic Symmetrization for Diverse Group Symmetries

Core Concepts

A novel framework for learning equivariant functions by symmetrizing an arbitrary base model using a learned equivariant distribution, which can handle diverse group symmetries including permutations, rotations, and their combinations.

Abstract

The paper presents a novel framework called "probabilistic symmetrization" for learning equivariant functions. The key idea is to replace the uniform distribution over the group in traditional group averaging with a parameterized equivariant distribution pω(g|x) that is trained end-to-end with the base model fθ. The main highlights are: Probabilistic symmetrization guarantees equivariance and universal approximation capability, while allowing the use of general-purpose architectures like MLP and transformer as the base model fθ. The equivariant distribution pω(g|x) is implemented as a noise-outsourced map that satisfies the condition of probabilistic equivariance, enabling gradient-based training. Implementations of pω(g|x) are provided for a wide range of practical symmetry groups including permutations (Sn), rotations (O(n), SO(n)), and their product combinations. Experiments show that probabilistic symmetrization achieves competitive or better performance compared to tailored equivariant architectures on various invariant and equivariant tasks, and can benefit from transferring knowledge from non-symmetric domains. The framework provides a general and flexible approach to learning equivariant functions, decoupling the group symmetry from the base model architecture.

Stats

The cardinality of the symmetry group G can be large or infinite, making exact group averaging intractable. For the n-body problem dataset, the base model fθ (a transformer) has around 2.3x more parameters than the baselines. For the PATTERN dataset, the equivariant distribution pω (a 3-layer GIN) has only 0.02% of the base model (pre-trained ViT) parameters.

Quotes

"We present a novel framework to overcome the limitations of equivariant architectures in learning functions with group symmetries." "Probabilistic symmetrization with equivariant distribution pω guarantees equivariance as well as expressive power of the symmetrized ϕθ,ω." "Empirical tests show competitive results against tailored equivariant architectures, suggesting the potential for learning equivariant functions for diverse groups using a non-equivariant universal base architecture."

Key Insights Distilled From

Learning Probabilistic Symmetrization for Architecture Agnostic Equivariance

by Jinwoo Kim,T... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2306.02866.pdf

Deeper Inquiries

How can the proposed probabilistic symmetrization framework be extended to handle continuous group symmetries, such as the special Euclidean group SE(3)

The extension of the probabilistic symmetrization framework to handle continuous group symmetries like the special Euclidean group SE(3) involves adapting the equivariant distribution pω(g|x) to account for the continuous nature of the group transformations. In the case of SE(3), which describes rotations and translations in 3D space, the distribution pω(g|x) needs to generate valid rotation matrices and translation vectors based on the input data x. To handle SE(3) symmetries, the equivariant network qω(x, ϵ) would need to output valid rotation matrices and translation vectors. This can be achieved by using appropriate parameterizations and constraints to ensure that the generated transformations are valid elements of SE(3). The noise variable ϵ can be sampled from a distribution that respects the invariances of SE(3), such as a normal distribution for rotations and translations. Additionally, postprocessing steps may be required to ensure that the output of the equivariant network corresponds to valid elements of SE(3). For example, the Gram-Schmidt process can be used to orthogonalize the rotation matrices to ensure they remain in the special orthogonal group SO(3). By adapting the probabilistic symmetrization framework in this way, it can effectively handle continuous group symmetries like SE(3) while maintaining equivariance and expressive power.

What are the potential limitations or drawbacks of the probabilistic approach compared to deterministic symmetrization methods, and how can they be addressed

One potential limitation of the probabilistic symmetrization approach compared to deterministic symmetrization methods is the increased computational cost associated with sampling from the equivariant distribution pω(g|x). Sampling multiple times during training and testing can lead to higher computational overhead, especially for complex or high-dimensional group symmetries. To address this limitation, several strategies can be employed: Efficient Sampling Techniques: Utilize efficient sampling techniques such as importance sampling or variance reduction methods to reduce the computational burden of sampling from the distribution. Approximate Inference: Implement approximate inference methods that provide a good approximation of the distribution without the need for exhaustive sampling. Parallelization: Utilize parallel computing resources to speed up the sampling process and reduce overall training time. Adaptive Sampling: Dynamically adjust the number of samples based on the complexity of the symmetry group or the specific task requirements to balance computational cost and performance. By implementing these strategies, the potential drawbacks of the probabilistic approach in terms of computational cost can be mitigated, making it more efficient and practical for a wide range of applications.

Can the insights from this work on transferring knowledge across symmetric and non-symmetric domains be applied to other areas of machine learning, such as few-shot learning or domain adaptation

The insights gained from transferring knowledge across symmetric and non-symmetric domains in the context of probabilistic symmetrization can be applied to other areas of machine learning, such as few-shot learning and domain adaptation. Few-Shot Learning: The idea of leveraging pre-trained models and transferring knowledge across different symmetries can be beneficial in few-shot learning scenarios. By pre-training on data from one domain and fine-tuning on a few examples from a related but different domain, the model can quickly adapt and generalize to new tasks with limited data. Domain Adaptation: The concept of transferring knowledge across symmetric and non-symmetric domains can also be applied to domain adaptation tasks. By learning invariant or equivariant representations in one domain and transferring this knowledge to another domain with different symmetries, the model can effectively adapt to new data distributions and improve generalization performance. By incorporating the principles of knowledge transfer and symmetry-aware learning from the probabilistic symmetrization framework, machine learning models can become more versatile, robust, and efficient in handling diverse datasets and tasks.