Probability-Signature Dynamics: Unpacking Modular Addition Learning Within Two-Layer Networks
Neural networks trained on modular addition tasks often develop Fourier-structured representations that support exact generalization. While prior work has identified these Fourier circuits, the mechanism by which gradient-based training selects them from the data distribution remains unclear. We address this question using probability signatures, which express leading gradient interactions through conditional statistics of the training distribution. For modular addition, these signatures are cyclic shift operators and are diagonalized by the discrete Fourier transform, yielding approximately decoupled Fourier-mode dynamics. This explains the emergence of Fourier sparsity, frequency matchi...
| Authors | Yunji Wang, Junjie Yao, Linyu Liu et al. |
|---|---|
| Primary category | cs.AI |
| Published | 8 October 2026 |
| Last updated at source | 8 October 2026 |
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