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Implicit neural networks are a general class of learning models that replacethe layers in traditional feedforward models with implicit algebraic equations.Compared to traditional learning models, implicit networks offer competitiveperformance and reduced memory consumption. However, they can remain brittlewith respect to input adversarial perturbations.  This paper proposes a theoretical and computational framework for robustnessverification of implicit neural networks; our framework blends together mixedmonotone systems theory and contraction theory. First, given an implicit neuralnetwork, we introduce a related embedded network and show that, given an$\ell_\infty$-norm box constraint on the input, the embedded network providesan $\ell_\infty$-norm box overapproximation for the output of the givennetwork. Second, using $\ell_{\infty}$-matrix measures, we propose sufficientconditions for well-posedness of both the original and embedded system anddesign an iterative algorithm to compute the $\ell_{\infty}$-norm boxrobustness margins for reachability and classification problems. Third, ofindependent value, we propose a novel relative classifier variable that leadsto tighter bounds on the certified adversarial robustness in classificationproblems. Finally, we perform numerical simulations on a Non-Euclidean MonotoneOperator Network (NEMON) trained on the MNIST dataset. In these simulations, wecompare the accuracy and run time of our mixed monotone contractive approachwith the existing robustness verification approaches in the literature forestimating the certified adversarial robustness.

Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approach