Statistical periodicity in noise-induced order from Ruelle-Pollicott resonances
arXiv:2607.18771
2026
Dynamics
1 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper identifies Ruelle–Pollicott resonances of an annealed transfer operator as the mechanism controlling statistical periodicity in noisy dynamical systems, independently of trajectory-level Lyapunov stability. A resonance z_j=r_j exp(i omega_j) predicts an observable-spectrum peak near omega_j and a correlation decay rate determined by r_j. This distinction can transfer to stochastic recurrent and state-space neural networks as a resonance monitor or regularizer, separating useful long-memory oscillations from unwanted ringing and exploding hidden-state dynamics.
Ideas from this paper
✓✓ Beats tuned baseline
2026
Estimate the leading complex resonances of the noise-averaged hidden-state dynamics of a stochastic RNN and use them to detect or control statistically persistent oscillations. The key design principle is to treat resonance radius and Lyapunov growth as independent signals: hidden trajectories can be Lyapunov-stable while the annealed dynamics still produce narrow-band ringing because a transfer-operator eigenvalue lies close to the unit circle.
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