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

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 \(\log 2-ξ\).

Disclosure

“22 LARGE DEVIATIONS FOR LINEAR REGIONS Declarations Acknowledgements. The authors used OpenAI’s ChatGPT during the preparation of this man- uscript. All mathematical statements, proofs, computations, references, and conclusions were in- dependently checked and verified by the authors, who take full responsibility for the content of the manus”

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Classification

Brainstorming or outlining

Multiplier

2

Verified

2026-08-24T00:34:15.228Z

Structural counts