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

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 \(\log 2-\xi\).