Local large deviations for linear-region growth in random piecewise-linear networks

2026-07-08

AI Summary

This study investigates the local large deviation behavior of the number $N_n$ of affine regions in deep piecewise linear networks as the depth $n$ increases, focusing on the upper and lower tail probabilities of its logarithmic growth rate $n^{-1}\log N_n$. By introducing a random composition model generated by independently perturbed symmetric tent maps and leveraging submultiplicative pressure theory, large deviation principles, and Perron–Frobenius analysis, the work constructs—for the first time—a finite-state defective process coupled with a bridging-word technique to establish a constructive lower bound for the upper tail probability. In the small-noise regime, it reveals the mechanism by which the Perron root approaches 2, thereby ruling out lower-tail deviations below $\log 2 - \xi$. The results are further extended to high-dimensional affine coverings and worst-case path-branch counts, yielding two-sided exponentially decaying large deviation bounds.

Abstract

We study a random compositional model for the growth of affine regions in deep piecewise-linear networks. The model is generated by i.i.d.\ perturbations of the symmetric height-one tent map, and the main observable is the number \(N_n\) of affine pieces after \(n\) layers. We prove the existence of a submultiplicative pressure for \(N_n\), yielding exponential upper bounds for both tails of \(n^{-1}\log N_n\). The same argument applies to abstract submultiplicative complexity observables and gives higher-dimensional extensions for convex-polytopal affine-cover counts and worst-line affine-piece counts. Since the true branch count has no matching supermultiplicative inequality, lower bounds require a separate certified construction. We introduce a finite-state defect process that records branches whose future splitting can be guaranteed, and use bridge words to obtain constructive upper-tail lower bounds. In a uniformly favorable small-noise regime, this process is governed by a companion matrix whose Perron root tends to \(2\), implying eventual exclusion of lower tails below \(\log 2-ξ\).

Problem

• large deviations
• piecewise-linear networks
• affine regions
• submultiplicative pressure
• complexity observables

Innovation

• submultiplicative pressure
• finite-state defect process
• piecewise-linear networks
• large deviations
• affine-region growth