Symmetry without a manifold: intrinsic dimension on orbits
The standard geometric derivation of neural scaling exponents takes the intrinsic dimension of a data manifold as its input. On modular addition in $\mathbb{Z}_p$ that derivation has no input. The exact algebraic solution is an orbit of $\mathbb{Z}_p$ acting by isometries. Transitivity alone makes the ratio statistic underlying the standard dimension estimator a point mass, so the estimator is undefined, and here the two nearest neighbour distances coincide exactly. Breaking the symmetry at scale $ε$ returns a number, but one that tracks $1/ε$ with no scale free plateau. We show that the failure is general, since on any finite orbit of a group acting by isometries the estimator reports th...
| Authors | Chon-Fai Kam, Miloud Bessafi, Frédéric Cadet |
|---|---|
| Primary category | cs.LG |
| Published | 15 September 2026 |
| Last updated at source | 15 September 2026 |
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