Partial Symmetry Enforced Attention Decomposition (PSEAD): A Group-Theoretic Framework for Equivariant Transformers in Biological Systems

Abstract: This research introduces the Theory of Partial Symmetry Enforced Attention Decomposition (PSEAD), a new and rigorous group-theoretic framework designed to seamlessly integrate local symmetry awareness into the core architecture of self-attention mechanisms within Transformer models. We formalize the concept of local permutation subgroup actions on windows of biological data, proving that under such actions, the attention mechanism naturally decomposes into a direct sum of orthogonal irreducible components. Critically, these components are intrinsically aligned with the irreducible representations of the acting permutation subgroup, thereby providing a powerful mathematical basis for disentangling symmetric and asymmetric features. We show that PSEAD offers substantial advantages. These include enhanced generalization capabilities to novel biological motifs exhibiting similar partial symmetries, unprecedented interpretability by allowing direct visualization and analysis of attention contributions from different symmetry channels, and significant computational efficiency gains by focusing representational capacity on relevant symmetric subspaces. Beyond static data analysis, we extend PSEAD's applicability to dynamic biological processes within reinforcement learning paradigms, showcasing its potential to accelerate the discovery and optimization of biologically meaningful policies in complex environments like protein folding and drug discovery. This work lays the groundwork for a new generation of biologically informed, symmetry-aware artificial intelligence models.