Robust Training and Verification of Implicit Neural Networks: A Non-Euclidean Contractive Approach

The paper’s Theorem 6.3 states four claims about an embedded implicit network, contraction of α-averaged iterations, and interval output bounds under the assumptions that Φ is monotone and 1-Lipschitz and that there exists η>0 with μ∞,[η]−1(W)<1; it also specifies the admissible range for α and refers elsewhere for the proof. The candidate solution reconstructs a detailed contraction-based proof using the weighted ∞-matrix measure, the Metzler/non-Metzler split, and mixed-monotone embedding, and derives precisely the items (1)–(4) in Theorem 6.3, including the role of α and the output bounds via [C]+/[C]−. This matches the paper’s statements and proof approach, which is explicitly grounded in non-Euclidean contraction and mixed-monotone embeddings (see the theorem statement and surrounding discussion). Therefore, the paper’s result and the candidate solution agree in content and technique; the paper omits the full proof in this PDF and cites prior work, while the candidate gives a complete proof in the same spirit.