A Unified Approach to Uniform Signal Recovery From Nonlinear Observations

Recent advances in quantized compressed sensing and high-dimensional estimation have shown that signal recovery is even feasible under strong nonlinear distortions in the observation process. An important characteristic of associated guarantees is uniformity, i.e., recovery succeeds for an entire class of structured signals with a fixed measurement ensemble. However, despite significant results in various special cases, a general understanding of uniform recovery from nonlinear observations is still missing. This paper develops a unified approach to this problem under the assumption of i.i.d. sub-Gaussian measurement vectors. Our main result shows that a simple least-squares estimator with any convex constraint can serve as a universal recovery strategy, which is outlier robust and does not require explicit knowledge of the underlying nonlinearity. Based on empirical process theory, a key technical novelty is an approximative increment condition that can be implemented for all common types of nonlinear models. This flexibility allows us to apply our approach to a variety of problems in nonlinear compressed sensing and high-dimensional statistics, leading to several new and improved guarantees. Each of these applications is accompanied by a conceptually simple and systematic proof, which does not rely on any deeper properties of the observation model. On the other hand, known local stability properties can be incorporated into our framework in a plug-and-play manner, thereby implying near-optimal error bounds.