Spectral Edge Dynamics Reveal Functional Modes of Learning
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914455187619840 |
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| author | Xu, Yongzhong |
| author_facet | Xu, Yongzhong |
| contents | 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 in the discrete-log basis, yielding a 5.9x improvement in concentration. For subtraction, the edge spans a small multi-mode family. For $x^2+y^2$, no single harmonic basis suffices, but cross-terms of additive and multiplicative features provide a 4x variance boost, consistent with the decomposition (a+b)^2 - 2ab. Multitask training amplifies this compositional structure, with the $x^2+y^2$ spectral edge inheriting the addition circuit's characteristic frequency (2.3x concentration increase).
These results suggest that training discovers low-dimensional functional modes over the input domain, whose structure depends on the algebraic symmetry of the task.
These results suggest that spectral edge dynamics identify low-dimensional functional subspaces governing learning, whose representation depends on the algebraic structure of the task. Simple harmonic structure emerges only when the task admits a symmetry-adapted basis; more complex tasks require richer functional descriptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06256 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Edge Dynamics Reveal Functional Modes of Learning Xu, Yongzhong Machine Learning Artificial Intelligence 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 in the discrete-log basis, yielding a 5.9x improvement in concentration. For subtraction, the edge spans a small multi-mode family. For $x^2+y^2$, no single harmonic basis suffices, but cross-terms of additive and multiplicative features provide a 4x variance boost, consistent with the decomposition (a+b)^2 - 2ab. Multitask training amplifies this compositional structure, with the $x^2+y^2$ spectral edge inheriting the addition circuit's characteristic frequency (2.3x concentration increase). These results suggest that training discovers low-dimensional functional modes over the input domain, whose structure depends on the algebraic symmetry of the task. These results suggest that spectral edge dynamics identify low-dimensional functional subspaces governing learning, whose representation depends on the algebraic structure of the task. Simple harmonic structure emerges only when the task admits a symmetry-adapted basis; more complex tasks require richer functional descriptions. |
| title | Spectral Edge Dynamics Reveal Functional Modes of Learning |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2604.06256 |