Jakob Paul Zimmermann · 2026 · arXiv (Cornell University)
DOI: 10.48550/arxiv.2610.26795 · cited by 0 works (OpenAlex) · open-access PDF
Setiap fungsi linear pecahan berkelanjutan (CPWL) adalah jumlah maksimum yang ditandatangani yang terbatas dari fungsi afine, membuat representasi maksimum menjadi masalah sentral dalam studi ekspresivitas jaringan ReLU. Sebuah perkiraan yang sekarang dibantah memprediksi kedalaman logaritma untuk maksimum. Terinspirasi oleh bukti oleh Brunck et al. (2026) bahwa dua lapisan tersembunyi cukup untuk fungsi CPWL umum, kami memberikan Caterpillar Net, sebuah konstruksi non-recursive untuk maksimum input $N$ dengan dua lapisan tersembunyi dari lebar $O(N^2)$ dan $2^O(N\Nlog)$. Konstruksi ini memiliki interpretasi geometris langsung.
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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