Spectral Edge Dynamics: An Analytical-Empirical Study of Phase Transitions in Neural Network Training
We develop the spectral edge analysis: phase transitions in neural network training -- grokking, capability gains, loss plateaus -- are controlled by the spectral gap of the rolling-window Gram matrix of parameter updates. In the extreme aspect ratio regime (parameters $P \sim 10^8$, window $W \sim 10$), the classical BBP detection threshold is vacuous; the operative structure is the intra-signal gap separating dominant from subdominant modes at position $k^* = \mathrm{argmax}\, σ_j/σ_{j+1}$. From three assumptions we derive: (i) gap dynamics governed by a Dyson-type ODE with curvature asymmetry, damping, and gradient driving; (ii) a spectral loss decomposition linking each mode's learnin...
| Authors | Yongzhong Xu |
|---|---|
| Primary category | cs.LG |
| Published | 30 March 2026 |
| Last updated at source | 7 May 2026 |
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