Understanding the great empirical success of artificial neural networks (NNs) from a theoretical point of view is currently one of the hottest research topics in computer science. In this paper we study the expressive power of NNs with rectified linear units from a combinatorial optimization perspective. In particular, we show that, given a directed graph with $n$ nodes and $m$ arcs, there exists an NN of polynomial size that computes a maximum flow from any possible real-valued arc capacities as input. To prove this, we develop the pseudo-code language Max-Affine Arithmetic Programs (MAAPs) and show equivalence between MAAPs and NNs concerning natural complexity measures. We then design a MAAP to exactly solve the Maximum Flow Problem, which translates to an NN of size $\mathcal{O}(m^2 n^2)$.

中文翻译：

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精确的最大流量计算的ReLU神经网络

从理论的角度理解人工神经网络（NNs）的巨大成功经验是当前计算机科学中最热门的研究主题之一。在本文中，我们从组合优化的角度研究了具有整流线性单元的神经网络的表达能力。特别地，我们表明，给定一个具有$ n $个节点和$ m $个弧的有向图，存在一个多项式大小的NN，该NN可从任何可能的实值弧容量作为输入来计算最大流量。为了证明这一点，我们开发了伪代码语言“最大仿射算术程序”（MAAP），并展示了关于自然复杂性度量的MAAP与NN之间的等效性。然后，我们设计一个MAAP来精确解决最大流量问题，这将转化为大小为\\ mathcal {O}（m ^ 2 n ^ 2）$的NN。