Title: Theoretical Constraints on the Expressive Power of RoPE-based Tensor Attention Transformers
Primary area: foundation or frontier models, including LLMs
Keywords: tensor attention, transformer, circuit complexity, machine learning theory

Abstract: Tensor Attention extends traditional attention mechanisms by capturing high-order correlations across multiple modalities, addressing the limitations of classical matrix-based attention. Meanwhile, Rotary Position Embedding ($\mathsf{RoPE}$) has shown superior performance in encoding positional information in long-context scenarios, significantly enhancing transformer models' expressiveness. Despite these empirical successes, the theoretical limitations of these technologies remain underexplored. In this study, we analyze the circuit complexity of Tensor Attention Transformer and extend to its $\mathsf{RoPE}$-based Tensor Attention variants, showing that with polynomial precision, constant-depth layers, and linear or sublinear hidden dimension, they cannot solve fixed membership problems or $(A_{F,r})^*$ closure problems, under the assumption that $\mathsf{TC}^0 \neq \mathsf{NC}^1$. These findings highlight a gap between the empirical performance and theoretical constraints of Tensor Attention and $\mathsf{RoPE}$-based Tensor Attention Transformers, offering insights that could guide the development of more theoretically grounded approaches to Transformer model design and scaling.