Quantitative Approximation Rates for Group Equivariant Learning
arXiv:2602.20370v1 Announce Type: new Abstract: The universal approximation theorem establishes that neural networks can approximate any continuous function on a compact set. Later works in approximation theory provide quantitative approximation rates for ReLU networks on the class of $\alpha$-H\"older functions $f: [0,1]^N \to \mathbb{R}$. The goal of this paper is to provide similar quantitative approximation results in the context of group equivariant learning, where the learned $\alpha$-H\"older function is known to obey certain group symmetries. While there has been much interest in the literature in understanding the universal approximation properties of equivariant models, very few quantitative approximation results are known for equivariant models. In this paper, we bridge this gap by deriving quantitative approximation rates for several prominent group-equivariant and invariant architectures. The architectures that we consider include: the permutation-invariant Deep Sets ar
arXiv:2602.20370v1 Announce Type: new Abstract: The universal approximation theorem establishes that neural networks can approximate any continuous function on a compact set. Later works in approximation theory provide quantitative approximation rates for ReLU networks on the class of $\alpha$-H\"older functions $f: [0,1]^N \to \mathbb{R}$. The goal of this paper is to provide similar quantitative approximation results in the context of group equivariant learning, where the learned $\alpha$-H\"older function is known to obey certain group symmetries. While there has been much interest in the literature in understanding the universal approximation properties of equivariant models, very few quantitative approximation results are known for equivariant models. In this paper, we bridge this gap by deriving quantitative approximation rates for several prominent group-equivariant and invariant architectures. The architectures that we consider include: the permutation-invariant Deep Sets architecture; the permutation-equivariant Sumformer and Transformer architectures; joint invariance to permutations and rigid motions using invariant networks based on frame averaging; and general bi-Lipschitz invariant models. Overall, we show that equally-sized ReLU MLPs and equivariant architectures are equally expressive over equivariant functions. Thus, hard-coding equivariance does not result in a loss of expressivity or approximation power in these models.
Executive Summary
This article bridges the gap in the field of group equivariant learning by deriving quantitative approximation rates for several prominent architectures. By comparing the expressiveness of ReLU MLPs and equivariant models, the authors demonstrate that hard-coding equivariance does not result in a loss of expressivity or approximation power. This work has significant implications for the development of machine learning models that can leverage symmetries in data, particularly in applications where equivariance is crucial such as computer vision and robotics. The results of this study provide a foundation for the design of more efficient and effective equivariant models, with the potential to improve performance in tasks such as object recognition and scene understanding.
Key Points
- ▸ Derivation of quantitative approximation rates for group-equivariant and invariant architectures
- ▸ Comparison of expressiveness between ReLU MLPs and equivariant models
- ▸ Demonstration of equal expressivity for equally-sized ReLU MLPs and equivariant architectures
Merits
Strength of Theory
The authors develop a rigorous theoretical framework for deriving quantitative approximation rates in the context of group equivariant learning, which is a significant advancement in the field.
Relevance to Applications
The results of this study have significant implications for the development of machine learning models that can leverage symmetries in data, particularly in applications where equivariance is crucial such as computer vision and robotics.
Demerits
Limited Scope
The study focuses on a limited set of architectures and may not be generalizable to other equivariant models or applications.
Experimental Validation
While the theoretical framework is developed, experimental validation of the results is not provided, which may limit the practical impact of the study.
Expert Commentary
The study provides a significant contribution to the field of equivariant learning by developing a rigorous theoretical framework for deriving quantitative approximation rates. The results of this study have implications for the development of machine learning models that can leverage symmetries in data, particularly in applications where equivariance is crucial. However, the study has limitations, such as a limited scope and a lack of experimental validation. Despite these limitations, the study provides a foundation for the design of more efficient and effective equivariant models, which can improve performance in tasks such as object recognition and scene understanding.
Recommendations
- ✓ Future studies should focus on experimental validation of the results and explore the generalizability of the theoretical framework to other equivariant models and applications.
- ✓ The results of this study should be considered when designing machine learning models that can leverage symmetries in data, particularly in applications where equivariance is crucial.
