Expanding Expressivity in Transformer Models with M\"obiusAttention

Attention mechanisms and Transformer architectures have revolutionized Natural Language Processing (NLP) by enabling exceptional modeling of long-range dependencies and capturing intricate linguistic patterns. However, their inherent reliance on linear operations in the form of matrix multiplications limits their ability to fully capture inter-token relationships on their own. We propose M\"obiusAttention, a novel approach that integrates M\"obius transformations within the attention mechanism of Transformer-based models. M\"obius transformations are non-linear operations in spaces over complex numbers with the ability to map between various geometries. By incorporating these properties, M\"obiusAttention empowers models to learn more intricate geometric relationships between tokens and capture a wider range of information through complex-valued weight vectors. We build and pre-train a BERT and a RoFormer version enhanced with M\"obiusAttention, which we then finetune on the GLUE benchmark. We evaluate empirically our approach against the baseline BERT and RoFormer models on a range of downstream tasks. Our approach compares favorably against the baseline models, even with smaller number of parameters suggesting the enhanced expressivity of M\"obiusAttention. This research paves the way for exploring the potential of M\"obius transformations in the complex projective space to enhance the expressivity and performance of foundation models.