Jakob Paul Zimmermann · 2026 · arXiv (Cornell University)
DOI: 10.48550/arxiv.2610.26795 · cited by 0 works (OpenAlex) · open-access PDF
Cada función lineal continua (CPWL) es una suma finita firmada de máximas de funciones afines, haciendo de la representación máxima un problema central en el estudio de la expresividad de las redes ReLU. Una conjetura ahora desafiada predice la profundidad logarítmica de las máximas. Inspirado en la prueba de Brunck et al. (2026) de que dos capas ocultas son suficientes para las funciones generales de CPWL, damos la Caterpillar Net, una construcción no recursiva para el máximo de entradas de $N$ con dos capas ocultas de anchos $O(N^2)$ y $2^O(N\Nlog)$. La construcción tiene una interpretación geométrica directa.
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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