Optimal Symmetries in Binary Classification

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.