揭示Transformer层中表征与因果能力的非线性差异，发现因果维度可量化且不随模型规模变化。
Causal Dimensionality of Transformer Representations: Measurement, Scaling, and Layer Structure
- 用SAE宽度扫描+归因修补法测量层间因果维度 kappa
- 表征容量增长15.6倍，因果容量仅增长4.35倍，存在显著分离
- 因果维度在不同模型和深度下保持稳定，适合研究模型内部机制
稀疏自编码器（SAEs）将Transformer残差流分解为可解释特征字典，但其宽度与对模型输出的因果影响之间的关系尚未系统刻画。本文引入因果维度 kappa(L, M, T)，定义为第L层期望雅可比外积的有效秩，并通过SAE宽度扫描结合归因修补法进行估计。在Gemma-2-2B第12层上，从16,384到1,048,576个特征的七种SAE宽度下，表征容量增长15.6倍，而因果容量仅增长4.35倍，形成稳健的‘表征-因果楔形’。饱和拟合得 kappa-hat ≈ 1,990，kappa-hat / d_model = 0.86，参与率下界 kappa_PR ≈ 280。关键的是，kappa在模型缩放下不变：尽管Gemma-2-9B参数量是Gemma-2-2B的3.46倍，但在相同SAE宽度下两者N_causal = 328（由校准强制为SAE宽度的2%；实质主张是AtP分数分布形状在序列长度=512时保持不变）。跨八层网络深度，kappa恒定，而绝对归因阈值从第1层到第23层下降20倍。五项控制实验（架构不变性、阈值鲁棒性、几何优势、合成真值恢复、四单元编码器/解码器消融）明确了kappa的度量内容与边界。研究结果确立了kappa作为Transformer层内生属性的可测性：其因果能力随SAE宽度亚线性恢复，对模型缩放不变，且在深度上具有结构化特征。
原文摘要 · Abstract (English)
Sparse autoencoders (SAEs) decompose transformer residual streams into interpretable feature dictionaries, yet the relationship between SAE width and causal influence on model output has not been systematically characterised. We introduce causal dimensionality kappa(L, M, T), defined as the effective rank of the expected Jacobian outer product at layer L, and show it can be estimated via the SAE width sweep paired with attribution patching. Across seven SAE widths from 16,384 to 1,048,576 features on Gemma-2-2B layer 12, representational capacity grows 15.6x while causal capacity grows only 4.35x: a robust separation we term the representational-causal wedge. A saturating fit yields kappa-hat approximately 1,990 with kappa-hat / d_model = 0.86 and participation-ratio lower bound kappa_PR approximately 280. Crucially, kappa is invariant to model scaling: Gemma-2-9B and Gemma-2-2B yield identical N_causal = 328 at the same SAE width despite a 3.46x parameter increase (the count is forced to 2% of SAE width by calibration; the substantive empirical claim is shape invariance of the AtP score distribution under matched seq=512 conditions). Across eight network depths kappa is constant while the absolute attribution threshold drops 20x from layer 1 to layer 23. Five controls (architecture invariance, threshold robustness, geometric privilege, synthetic ground-truth recovery, and a four-cell encoder/decoder ablation) pin down what kappa measures and what it does not. Our findings establish kappa as a measurable, model-intrinsic property of transformer layers: sub-linearly recoverable by SAE width, invariant to model scaling, and structured across network depth.
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