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Bridging the Dimensional Chasm: Uncover Layer-wise Dimensional Reduction in Transformers through Token Correlation

The geometric evolution of token representations in large language models (LLMs) presents a fundamental paradox: while human language inherently organizes semantic information in low-dimensional spaces ($\sim 10^1$ dimensions), modern LLMs employ high-dimensional embeddings ($\sim 10^3$ dimensions) processed through Transformer architectures. To resolve this paradox, this work bridges this conceptual gap by developing a geometric framework that tracks token dynamics across Transformers layers. Through layer-wise analysis of intrinsic dimensions across multiple architectures, we reveal an expansion-contraction pattern where tokens diffuse to a "working space" and then progressively project onto lower-dimensional submanifolds. Our finding implies a negative correlation between the working space dimension and parameter-sensitive performance of the LLMs, and indicates that effective models tend to compress tokens into approximately 10-dimensional submanifolds, closely resembling human semantic spaces. This work not only advances LLM interpretability by reframing Transformers layers as projectors that mediate between high-dimensional computation and low-dimensional semantics, but also provides practical tools for model diagnostics that do not rely on task-specific evaluations.

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