The Memory Hidden in Response Fluctuations: Trajectory-Level Fluctuation-Response Theory and Inequalities for Non-Markovian Jump Dynamics
arXiv:2608.20328
2026
Dynamics
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper develops an exact trajectory-level decomposition for history-dependent event dynamics: every event increment is split into a predictable intensity and an orthogonal martingale residual. Its key transferable asset is the event-consequence kernel, which measures how a particular past event changes a future observable, together with a nonnegative response-heterogeneity gap that tests whether a compressed memory representation preserves dynamically relevant information. In neural networks, this can provide a principled memory-compression criterion for RNNs or state-space models and a response-weighted variance controller for stochastic optimization. The central falsifiable signature is that response-sufficient compressed states have low conditional variance of event-consequence kernels, even when ordinary autocorrelation-based memory measures remain large.
Ideas from this paper
✗ Failed on benchmark
2026
Replace correlation-based memory pruning in an RNN or state-space model by measuring how hidden-state history changes the response to individual past input events. Train a compressed memory coordinate only if it preserves the event-consequence kernel for the target observable, such as future loss, prediction, or control return. A memory representation is accepted when the conditional variance of this kernel within compressed-state groups is small, even if dwell-time or autocorrelation…
Useful8/10
Difficulty6/10
Novelty8/10
✗ Failed on benchmark
2026
Use the trajectory martingale decomposition to separate predictable training updates from genuinely unpredictable residual updates, then scale the residual according to its estimated response to future loss. The method targets stochastic or event-driven optimization with history-dependent samples and predicts that response-weighted residual energy, rather than total gradient variance, controls update noise and instability.
Useful7/10
Difficulty7/10
Novelty7/10