Jakob Paul Zimmermann · 2026 · arXiv (Cornell University)
DOI: 10.48550/arxiv.2610.26795 · cited by 0 works (OpenAlex) · open-access PDF
Cada função linear fragmentada contínua (CPWL) é uma soma assinada finita de máximas de funções afinas, tornando a representação máxima um problema central no estudo da expressividade das redes ReLU. Uma conjectura agora desafiada previu profundidade logarítmica para máximas. Inspirado na prova de Brunck et al. (2026) de que duas camadas ocultas são suficientes para funções CPWL gerais, damos a Caterpillar Net, uma construção não recursiva para o máximo de $N$ entradas com duas camadas ocultas de larguras $O(N^2)$ e $2^O(N\Nlog)$. A construção tem uma interpretação geométrica direta.
Machine translation (Meta M2M100); the English original follows.
Every continuous piecewise linear (CPWL) function is a finite signed sum of maxima of affine functions, making maximum representation a central problem in the study of the expressiveness of ReLU networks. A now-disproved conjecture predicted logarithmic depth for maxima. Inspired by the proof by Brunck et al. (2026) that two hidden layers suffice for general CPWL functions, we give the Caterpillar Net, a nonrecursive construction for the maximum of $N$ inputs with two hidden layers of widths $O(N^2)$ and $2^{O(N\log N)}$. The construction has a direct geometric interpretation.
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Topics: Mathematics, Conjecture, Logarithm, Maxima, Affine transformation, Piecewise
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