A Relative Variational Principle for Expanding Iterated Function Systems
arXiv:2608.18426
2026
Dynamics
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper develops a relative variational principle for expanding, possibly nonstationary iterated-function systems: conditional entropy in the fiber, conditioned on an ergodic symbolic base, equals the base-average of fiberwise topological entropy. Its transferable asset is an entropy and free-energy accounting rule for dynamically changing multi-branch maps. A neural analogue is a routed or mixture-of-experts network whose branch transformations have measurable expansion rates and whose routing entropy is calibrated to fiber trajectory growth. This yields a falsifiable mechanism for preventing premature expert collapse while preserving specialization.
Ideas from this paper
Unverified
2026
Treat the router state as a symbolic base process and expert transformations as nonstationary expanding fiber maps. Add a relative entropy/free-energy constraint so that the router's conditional entropy is calibrated against the empirically measured growth rate of distinguishable expert trajectories, preventing premature expert collapse while retaining useful specialization.
Useful6/10
Difficulty6/10
Novelty7/10
Unverified
2026
Use a pressure objective to select expert-routing distributions by balancing task reward against route entropy, rather than optimizing task loss alone. The resulting router behaves like an equilibrium-state estimator: it should retain multiple high-performing branches when their combined entropy outweighs the advantage of a single branch.
Useful5/10
Difficulty5/10
Novelty5/10