A domain-theoretic framework is presented for validated robustness analysis of neural networks. First, global robustness of a general class of networks is analyzed. Then, using the fact that Edalat’s domain-theoretic L-derivative coincides with Clarke’s generalized gradient, the framework is extended for attack-agnostic local robustness analysis. The proposed framework is ideal for designing algorithms which are correct by construction. This claim is exemplified by developing a validated algorithm for estimation of Lipschitz constant of feedforward regressors. The completeness of the algorithm is proved over differentiable networks and also over general position ${\mathrm{ReLU}}$ networks. Computability results are obtained within the framework of effectively given domains. Using the proposed domain model, differentiable and non-differentiable networks can be analyzed uniformly. The validated algorithm is implemented using arbitrary-precision interval arithmetic, and the results of some experiments are presented. The software implementation is truly validated, as it handles floating-point errors as well.

We show that any Lipschitz projection-valued function $p$ on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions $q$ with Lipschitz constant close to that of $p$. This answers a question of Rieffel.

Let (X, τ) be a metrizable space and {fn: n = 1,2,…} be a commuting family of continuous mappings on X with a common fixed point ξ ∈ X such that (I) for each k = l, 2, … and (II) Then for each c ∈ (0,1), there exists a metric d on X inducing the topology τ such that , for all x, y∈X and n = l, 2, …. The above result is also generalized to Tychonoff spaces.