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 functions 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 the main theoretical results are verified in numerical simulations using the Modified National Institute of Standards and Technology (MNIST) dataset.

翻译：我们运用不动点理论分析非负神经网络——即映射非负向量到非负向量的神经网络。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）可伸缩函数。这一事实使我们能够为非负神经网络（其输入与输出维度相同）的不动点存在性提供条件，且这些条件弱于近期利用凸分析论证所获得的结果。此外，我们证明了具有非负权重和偏置的非负神经网络的不动点集呈区间形状，在温和条件下该区间退化为单点。这些结果随后被用于推导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行为特征，主要理论结果已通过使用修改版国家标准与技术研究所（MNIST）数据集的数值模拟得到验证。