{"ID":6138088,"CreatedAt":"2026-07-09T01:07:32.349475501Z","UpdatedAt":"2026-07-11T05:01:04.438793932Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.07014","arxiv_id":"2607.07014","title":"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\\).","short_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...","url_abs":"https://arxiv.org/abs/2607.07014","url_pdf":"https://arxiv.org/pdf/2607.07014v1","authors":"[\"Recep Özkan\",\"Christian Hirsch\"]","published":"2026-07-08T05:14:19Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"stat.ML\"]","methods":"[]","has_code":false}