We use fixed point theory to analyze nonnegative neural networks, which we define as neural networks that map nonnegative vectors to nonnegative vectors. We first show that nonnegative neural networks with nonnegative weights and biases can be recognized as monotonic and (weakly) scalable mappings within the framework of nonlinear Perron-Frobenius theory. This fact enables us to provide conditions for the existence of fixed points of nonnegative neural networks having inputs and outputs of the same dimension, and these conditions are weaker than those recently obtained using arguments in convex analysis. Furthermore, we prove that the shape of the fixed point set of nonnegative neural networks with nonnegative weights and biases is an interval, which under mild conditions degenerates to a point. These results are then used to obtain the existence of fixed points of more general nonnegative neural networks. From a practical perspective, our results contribute to the understanding of the behavior of autoencoders, and we also offer valuable mathematical machinery for future developments in deep equilibrium models.

翻译：我们利用不动点理论分析非负神经网络，这类神经网络定义为将非负向量映射到非负向量的网络。首先证明，具有非负权重和偏置的非负神经网络可在非线性Perron-Frobenius理论框架下被识别为单调且（弱）可缩映射。这一事实使我们能够为输入输出维度相同的非负神经网络提供不动点存在条件，且这些条件比近期使用凸分析论证得到的结果更弱。进一步证明，具有非负权重和偏置的非负神经网络的不动点集形状为区间，在温和条件下退化为单点。这些结果随后被用于获取更一般非负神经网络的不动点存在性。从实践角度而言，我们的研究有助于理解自编码器的运行行为，同时也为深度均衡模型的未来发展提供了有价值的数学工具。