Spectral Edge Dynamics Reveal Functional Modes of Learning
Training dynamics during grokking concentrate along a small number of dominant update directions -- the spectral edge -- which reliably distinguishes grokking from non-grokking regimes. We show that standard mechanistic interpretability tools (head attribution, activation probing, sparse autoencoders) fail to capture these directions: their structure is not localized in parameter or feature space. Instead, each direction induces a structured function over the input domain, revealing low-dimensional functional modes invisible to representation-level analysis. For modular addition, all leading directions collapse to a single Fourier mode. For multiplication, the same collapse appears only i...
| Authors | Yongzhong Xu |
|---|---|
| Primary category | cs.LG |
| Published | 6 April 2026 |
| Last updated at source | 6 April 2026 |
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