Jakob Paul Zimmermann · 2026 · arXiv (Cornell University)
DOI: 10.48550/arxiv.2610.26795 · cited by 0 works (OpenAlex) · open-access PDF
每一个连续的片面线性(CPWL)函数都是相似函数的最大值的有限签名总和,使最大表示成为研究 ReLU 网络的表达性的一个核心问题。 一个现在被驳回的猜测预测了最大值的逻辑深度。 灵感来自于 Brunck et al.(2026)的证据,即两个隐藏的层对一般 CPWL 函数来说是足够的,我们给了 Caterpillar Net,一个非重复性的结构,用于最大 $N$输入,其中有两个宽度 $O(N^2)$和 $2^O(N\Nlog)$的隐藏层。
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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