Scale-Consistent State-Space Dynamics via Fractal of Stationary Transformations

TL;DR

This paper introduces FROST, a fractal-based approach to enforce scale-consistent dynamics in state-space models, enabling stable iterative refinement and adaptive computation, validated through experiments on ImageNet-100.

Contribution

We propose FROST, a novel fractal inductive bias that enforces scale-consistent latent dynamics in state-space models, improving stability and enabling adaptive halting.

Findings

FROST enforces self-similar representation manifolds.

Stable convergence across iterations is theoretically established.

Experiments on ImageNet-100 confirm scale-consistent behavior and adaptive efficiency.

Abstract

Recent deep learning models increasingly rely on depth without structural guarantees on the validity of intermediate representations, rendering early stopping and adaptive computation ill-posed. We address this limitation by formulating a structural requirement for state-space model's scale-consistent latent dynamics across iterative refinement, and derive Fractal of Stationary Transformations (FROST), which enforces a self-similar representation manifold through a fractal inductive bias. Under this geometry, intermediate states correspond to different resolutions of a shared representation, and we provide a geometric analysis establishing contraction and stable convergence across iterations. As a consequence of this scale-consistent structure, halting naturally admits a ranking-based formulation driven by intrinsic feature quality rather than extrinsic objectives. Controlled experiments on ImageNet-100 empirically verify the predicted scale-consistent behavior, showing that adaptive efficiency emerges from the aligned latent geometry.