Optimal Symmetries in Binary Classification
Vishal S. Ngairangbam, Michael Spannowsky
TL;DR
The paper develops a fibre-based, stabiliser-centric framework for designing group-equivariant neural networks in binary classification, linking symmetry choices to the likelihood-ratio structure via Neyman–Pearson optimality. It argues that the optimal symmetry is not necessarily the largest group, but one whose invariance or equivariance preserves the fibre decomposition of the likelihood ratio, with clear implications for sample efficiency and generalisation. Theoretical results connect invariance/equivariance with fibre size and stabilisers, and experiments on 3D point-cloud tasks demonstrate that smaller, problem-aligned groups (e.g., $ ext{O}(2)$) can outperform larger symmetry groups like $ ext{E}(3)$ or $ ext{O}(3)$. The work provides unified guidelines for constructing symmetry-aware architectures and suggests future extensions to more complex domains and real-world noise robustness.
Abstract
We explore the role of group symmetries in binary classification tasks, presenting a novel framework that leverages the principles of Neyman-Pearson optimality. Contrary to the common intuition that larger symmetry groups lead to improved classification performance, our findings show that selecting the appropriate group symmetries is crucial for optimising generalisation and sample efficiency. We develop a theoretical foundation for designing group equivariant neural networks that align the choice of symmetries with the underlying probability distributions of the data. Our approach provides a unified methodology for improving classification accuracy across a broad range of applications by carefully tailoring the symmetry group to the specific characteristics of the problem. Theoretical analysis and experimental results demonstrate that optimal classification performance is not always associated with the largest equivariant groups possible in the domain, even when the likelihood ratio is invariant under one of its proper subgroups, but rather with those subgroups themselves. This work offers insights and practical guidelines for constructing more effective group equivariant architectures in diverse machine-learning contexts.