Implicit-depth neural networks have grown as powerful alternatives to traditional networks in various applications in recent years. However, these models often lack guarantees of existence and uniqueness, raising stability, performance, and reproducibility issues. In this paper, we present a new analysis of the existence and uniqueness of fixed points for implicit-depth neural networks based on the concept of subhomogeneous operators and the nonlinear Perron-Frobenius theory. Compared to previous similar analyses, our theory allows for weaker assumptions on the parameter matrices, thus yielding a more flexible framework for well-defined implicit networks. We illustrate the performance of the resulting subhomogeneous networks on feed-forward, convolutional, and graph neural network examples.

翻译：隐式深度神经网络近年来在各种应用中作为传统网络的有力替代方案而发展迅速。然而，这类模型通常缺乏存在性和唯一性的保证，从而引发稳定性、性能和可重复性问题。本文基于亚齐次算子概念和非线性佩龙-弗罗贝尼乌斯理论，对隐式深度神经网络不动点的存在性和唯一性提出了新的分析。与以往类似分析相比，我们的理论允许对参数矩阵施加更弱的假设条件，从而为良定义的隐式网络构建了更灵活的框架。我们通过前馈、卷积和图神经网络示例展示了所提出的亚齐次网络的性能。