Measuring Layer-wise Intrinsic Dimensionality of FFNs in LLMs via PCA

Modern large language models keep the feed-forward network (FFN) intermediate width identical across every layer, a convention inherited from the original Transformer. Yet mechanistic work suggests layers at different depths do qualitatively different work, raising a natural question: are their capacity demands also non-uniform, and if so, can a fixed parameter budget be allocated more wisely than uniformly? To probe this, we apply principal component analysis (PCA) to the input of each FFN down-projection and measure, for every layer, the smallest number of principal components needed to reconstruct the activation up to a fixed cosine similarity --- the layer's intrinsic dimensionality. Scanning 58 open-weight LLMs spanning 14 families and 70M to 72B parameters, we find a recurring middle-heavy profile in which middle layers carry higher intrinsic dimensionality than edge layers; the profile is robust to the calibration corpus, even across English and Chinese, and is largely preserved through post-training fine-tuning. As a direct test that this signal is structurally meaningful, we use it to guide width pruning in a knowledge-distilled student under a matched parameter budget: the PCA-guided allocation outperforms the uniform-width convention by $+2.7$ points on Qwen2.5-3B and $+2.3$ on Qwen3-8B, while a same-scale teacher (LLaMA-3-8B) shows only $+0.9$, and a budget-matched inverted control consistently does worst. Across these three teachers, the size of the gap scales with how unevenly the teacher's intrinsic dimensionality is distributed across layers --- a single-number summary we call $\sigma_{99}$ --- and the advantage persists through downstream supervised fine-tuning.