Algebraic Representability as the Limiting Regime of Grokking: An Exactly Solvable Model with Holomorphic Activations
Neural networks trained on modular arithmetic exhibit grokking, a delayed transition from memorisation to generalisation known to depend on model capacity: too little and the network memorises slowly or not at all, too much and it generalises almost immediately. What happens at the extreme of this spectrum, when the architecture's expressible function class collapses to a finite-dimensional algebraic variety? We study two-layer networks with a holomorphic monomial activation sigma(z)=z^k, trained on modular tasks encoded via roots of unity. Here the network output, regardless of hidden width, is confined to a (k+1)-dimensional subspace of characters of (Z_p)^2, an O(k/p^2) slice of the fu...
| Authors | Chon-Fai Kam, Xavier Cadet, Miloud Bessafi et al. |
|---|---|
| Primary category | cs.LG |
| Published | 15 July 2026 |
| Last updated at source | 15 July 2026 |
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