Neural networks with ReLU activation play a key role in modern machine learning. In view of safety-critical applications, the verification of trained networks is of great importance and necessitates a thorough understanding of essential properties of the function computed by a ReLU network, including characteristics like injectivity and surjectivity. Recently, Puthawala et al. [JMLR 2022] came up with a characterization for injectivity of a ReLU layer, which implies an exponential time algorithm. However, the exact computational complexity of deciding injectivity remained open. We answer this question by proving coNP-completeness of deciding injectivity of a ReLU layer. On the positive side, as our main result, we present a parameterized algorithm which yields fixed-parameter tractability of the problem with respect to the input dimension. In addition, we also characterize surjectivity for two-layer ReLU networks with one-dimensional output. Remarkably, the decision problem turns out to be the complement of a basic network verification task. We prove NP-hardness for surjectivity, implying a stronger hardness result than previously known for the network verification problem. Finally, we reveal interesting connections to computational convexity by formulating the surjectivity problem as a zonotope containment problem

翻译：采用ReLU激活函数的神经网络在现代机器学习中发挥着关键作用。鉴于安全关键型应用的需求，对训练后网络的验证至关重要，这需要深入理解ReLU网络所计算函数的基本性质，包括单射性和满射性等特征。最近，Puthawala等人[JMLR 2022]提出了ReLU网络层单射性的判定条件，该条件蕴含指数时间算法。然而，判定单射性的精确计算复杂性始终是未解难题。我们通过证明ReLU网络层单射性判定属于coNP完全问题来解答这一疑问。在积极方面，作为主要研究成果，我们提出了一种参数化算法，使得该问题在输入维度参数下具有固定参数可处理性。此外，我们还刻画了具有一维输出的双层ReLU网络的满射性特征。值得注意的是，该判定问题恰好构成基础网络验证任务的补集。我们证明了满射性判定具有NP困难性，这为网络验证问题提供了比以往已知结果更强的困难性证明。最后，通过将满射性问题表述为带状多面体包含问题，我们揭示了与计算凸性之间的有趣联系。