Jakob Paul Zimmermann · 2026 · arXiv (Cornell University)
DOI: 10.48550/arxiv.2610.26795 · cited by 0 works (OpenAlex) · open-access PDF
각 연속 조각형 선형 (CPWL) 함수는 친밀한 함수의 최대의 유한 서명 합계이며, 최대 표현을 ReLU 네트워크의 표현성 연구의 핵심 문제로 만듭니다. 현재 증명되지 않은 추측은 최대에 대한 로고리즘 깊이를 예측합니다. Brunck et al. (2026)의 두 개의 숨겨진 층이 일반적인 CPWL 함수에 충분하다는 증거에 영감을 받았으며, 우리는 $O(N^2)$와 $2^O(N\Nlog)$의 두 개의 숨겨진 층으로 $N$ 입력의 최대에 대한 비 회귀 구조 인 Caterpillar Net를 제공합니다.
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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