Unpacking Positional Encoding in Transformers: A Spectral Analysis of Content-Position Coupling

• URL: http://arxiv.org/abs/2505.13027v1
• Date: Mon, 19 May 2025 12:11:13 GMT
• Title: Unpacking Positional Encoding in Transformers: A Spectral Analysis of Content-Position Coupling
• Authors: Zihan Gu, Han Zhang, Ruoyu Chen, Yue Hu, Hua Zhang,
• Abstract summary: Positional encoding (PE) is essential for enabling Transformers to model sequential structure.<n>We present a unified framework that analyzes PE through the spectral properties of Toeplitz and related matrices.<n>We establish explicit content-relative mixing with relative-position Toeplitz signals as a key principle for effective PE design.
• Score: 10.931433906211534
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Positional encoding (PE) is essential for enabling Transformers to model sequential structure. However, the mechanisms by which different PE schemes couple token content and positional information-and how these mechanisms influence model dynamics-remain theoretically underexplored. In this work, we present a unified framework that analyzes PE through the spectral properties of Toeplitz and related matrices derived from attention logits. We show that multiplicative content-position coupling-exemplified by Rotary Positional Encoding (RoPE) via a Hadamard product with a Toeplitz matrix-induces spectral contraction, which theoretically improves optimization stability and efficiency. Guided by this theory, we construct synthetic tasks that contrast content-position dependent and content-position independent settings, and evaluate a range of PE methods. Our experiments reveal strong alignment with theory: RoPE consistently outperforms other methods on position-sensitive tasks and induces "single-head deposit" patterns in early layers, indicating localized positional processing. Further analyses show that modifying the method and timing of PE coupling, such as MLA in Deepseek-V3, can effectively mitigate this concentration. These results establish explicit content-relative mixing with relative-position Toeplitz signals as a key principle for effective PE design and provide new insight into how positional structure is integrated in Transformer architectures.

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