# Expanding Expressivity in Transformer Models with M"obiusAttention

arXiv: https://arxiv.org/abs/2409.12175 · HTML: https://mlsanity.com/paper/2409.12175/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 69 references that are in the map.
- First posted on arXiv 8 Sep 2024.

## Builds on (most cited first, 30 of 49)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [SQuAD: 100,000+ Questions for Machine Comprehension of Text](https://mlsanity.com/paper/1606.05250/) (2016)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [XLNet: Generalized Autoregressive Pretraining for Language Understanding](https://mlsanity.com/paper/1906.08237/) (2019)
- [ALBERT: A Lite BERT for Self-supervised Learning of Language Representations](https://mlsanity.com/paper/1909.11942/) (2019)
- [A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference](https://mlsanity.com/paper/1704.05426/) (2017)
- [DeBERTa: Decoding-enhanced BERT with Disentangled Attention](https://mlsanity.com/paper/2006.03654/) (2020)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Generating Long Sequences with Sparse Transformers](https://mlsanity.com/paper/1904.10509/) (2019)
- [Cross-lingual Language Model Pretraining](https://mlsanity.com/paper/1901.07291/) (2019)
- [Aligning Books and Movies: Towards Story-like Visual Explanations by Watching Movies and Reading Books](https://mlsanity.com/paper/1506.06724/) (2015)
- [Neural Network Acceptability Judgments](https://mlsanity.com/paper/1805.12471/) (2018)
- [SpanBERT: Improving Pre-training by Representing and Predicting Spans](https://mlsanity.com/paper/1907.10529/) (2019)
- [GPT-NeoX-20B: An Open-Source Autoregressive Language Model](https://mlsanity.com/paper/2204.06745/) (2022)

## Built on by (most cited first, 4 of 4)

- [Invertible Query-Key Coupling Composes with Attention Mechanisms](https://mlsanity.com/paper/2604.01683/) (2026)
- [Neural Attention: A Novel Mechanism for Enhanced Expressive Power in Transformer Models](https://mlsanity.com/paper/2502.17206/) (2025)
- [Anti-Periodic Positional Encoding: M"obius Boundary Conditions Make In-Context Retrieval Reliable](https://mlsanity.com/paper/2607.21405/) (2026)
- [Riccati State Space Models: Non-iterative Parallelization for Nonlinear Sequence Modeling](https://mlsanity.com/paper/2609.35441/) (2026)

## Abstract (arXiv)

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.