模型越大，表征几何越能保持预测能力的组织结构。
Scale Determines Whether Language Models Organize Representation Geometry for Prediction
- 用子空间PGA检测层间距离是否对齐解码矩阵的读出子空间。
- 小模型（d≤1024）在训练后期逐渐失去预测组织性，大模型（d≥2048）则持续保持。
- 发现容量限制导致少数主方向偏离解码方向，影响几何组织性。
在语言模型中，表征的意义由其表征空间的几何结构决定：距离承载信息，而非激活值。现有工具仅描述几何形状，未探讨其组织目的。本文提出子空间PGA，用于检验某层的距离结构是否比同尺寸随机子空间更贴近解码矩阵 $W_U$ 的读出子空间。在七组Pythia模型（70M–6.9B）及三组跨家族模型中，中间层几何显著服务于预测（最大 $z = 9$–$24$），但该组织性随规模变化：小模型（$d \≤ 1024$）在训练后期虽损失持续下降仍逐步丧失组织性；而大模型（$d \≥ 2048$）则全程维持。根源在于容量权衡——少数主导方向移出 $W_U$ 读出空间，掩盖而非破坏预测结构，去除这些方向可恢复对齐。谱特征与损失曲线均无法捕捉此差异。因此，规模不仅影响预测性能，也决定了表征几何如何被组织以支持预测。
原文摘要 · Abstract (English)
In language models, what a representation encodes is determined by the geometry of its representation space: distances, not activations, carry meaning. Existing tools characterize the shape of this geometry but do not ask what that shape is organized for. We introduce Subspace PGA, a metric that tests whether a layer's distance structure aligns with the readout subspace of the unembedding matrix $W_U$ more than with random subspaces of equal size. Across seven Pythia models (70M--6.9B) and three cross-family models, intermediate geometry is significantly organized for prediction (peak $z = 9$--$24$), but the degree is scale-dependent: small models ($d \leq 1024$) progressively lose it at late layers during training -- even as loss keeps improving -- while large models ($d \geq 2048$) preserve it throughout. We trace this to a capacity trade-off: a few dominant directions migrate away from $W_U$'s readout, masking rather than destroying the predictive structure beneath, and removing them restores alignment. Neither spectral metrics nor loss curves capture this distinction. Scale thus determines not only how well a model predicts, but how its representation geometry is organized to do so.
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