Kolmogorov--Arnold against bounded translations

Historically originating from Hilbert's 13th problem, the Kolmogorov-Arnold representation theorem (KART) has recently experienced a major revitalisation through its applications to neural networks, specifically Kolmogorov-Arnold Networks (KANs). While the exact representation is well established, its stability under continuous adversarial perturbations of the hidden layer remains a critical open question. In this paper, we investigate the robustness of KART against bounded adversarial translations. We provide an explicit, self-contained, and constructive proof of an approximate representation using fixed, piecewise linear inner functions. Crucially, our construction employs a single outer function that remains invariant for all summands and is independent of the specific adversarial translation, provided its maximum bound is known a priori.