Theoretical Constraints on the Expressive Power of $\mathsf{RoPE}$-based Tensor Attention Transformers

• URL: http://arxiv.org/abs/2412.18040v1
• Date: Mon, 23 Dec 2024 23:26:07 GMT
• Title: Theoretical Constraints on the Expressive Power of $\mathsf{RoPE}$-based Tensor Attention Transformers
• Authors: Xiaoyu Li, Yingyu Liang, Zhenmei Shi, Zhao Song, Mingda Wan,
• Abstract summary: We analyze the circuit complexity of Attention and $mathsfRoPE$-based Attention, showing that they cannot solve fixed membership problems or $(A_F,r)*$ closure problems. These findings highlight a gap between the empirical performance and theoretical constraints of Attention and $mathsfRoPE$-based Attention Transformers.
• Score: 23.991344681741058
• License:
• 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 and $\mathsf{RoPE}$-based Tensor Attention, 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.

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