Optimal lower Lipschitz bounds for ReLU layers, saturation, and phase retrieval

Abstract: The injectivity of ReLU layers in neural networks, the recovery of vectors from clipped or saturated measurements, and (real) phase retrieval in $\mathbb{R}^n$ allow for a similar problem formulation and characterization using frame theory. In this paper, we revisit all three problems with a unified perspective and derive lower Lipschitz bounds for ReLU layers and clipping which are analogous to the previously known result for phase retrieval and are optimal up to a constant factor.