  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.
