Tropical Circuits with Scalar Multiplication Gates

Christoph Hertrich; Moritz Stargalla

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arXiv · 2607.11540

Tropical Circuits with Scalar Multiplication Gates

Christoph Hertrich·Moritz Stargalla

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Abstract

We study tropical circuits with scalar multiplication gates, that is, algebraic circuits whose gates implement $\max$, $+$, or multiplication with a positive constant. For such circuits, we prove exponential size lower bounds for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings. As a corollary, we obtain an exponential size separation between monotone and non-monotone maxout neural networks, which generalize the popularly used ReLU neural networks. One conclusion from this is that neural network models with enforced convexity constraints, such as input-convex neural networks (ICNNs), sometimes need to be exponentially larger than their unrestricted counterparts in order to express the same functions.

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Christoph Hertrich, Moritz Stargalla. 2026-07-13. Tropical Circuits with Scalar Multiplication Gates. https://arxiv.org/abs/2607.11540

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