Annealed Ruelle-Pollicott Resonances
arXiv:2608.05649
2026
Dynamics
1 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper transfers Ruelle–Pollicott resonance theory from deterministic maps to iid random dynamical systems by replacing a single transfer or Koopman operator with an annealed operator obtained by averaging over random maps. Its key mechanism is that eigenvalues of this operator quantitatively determine the decay of annealed correlations, including oscillatory memory when resonances are complex. This can be transferred to stochastic recurrent networks or state-space models by estimating the Koopman operator on hidden-state features, constraining its nontrivial spectrum, and using resonance locations to design or monitor long-horizon memory.
Ideas from this paper
△ Mechanism confirmed, baseline not beaten
2026
Replace a deterministic recurrent transition by an iid-random family of transitions and explicitly control the spectrum of the corresponding annealed Koopman operator. Nontrivial eigenvalues inside the unit disk give a measurable exponential memory-decay envelope, while complex eigenvalues provide stable oscillatory memory modes useful for long-horizon sequence prediction.
Useful8/10
Difficulty6/10
Novelty7/10