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 feedforward, convolutional, and graph neural network examples.

翻译：近年来，隐式深度神经网络作为传统网络的有力替代方案，在各种应用中不断发展。然而，这些模型通常缺乏存在性和唯一性保证，引发了稳定性、性能与可复现性方面的问题。本文基于次齐次算子概念和非线性Perron-Frobenius理论，对隐式深度神经网络不动点的存在性与唯一性提出了新的分析框架。与先前类似分析相比，我们的理论允许对参数矩阵采用更弱的假设条件，从而为良定义的隐式网络提供了更灵活的框架。我们在前馈神经网络、卷积神经网络和图神经网络示例中展示了所构建的次齐次网络的性能表现。