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.

量化压缩感知和高维估计的最新进展表明，在观测过程中的强非线性失真下，信号恢复甚至是可行的。相关保证的一个重要特征是一致性，即，对于具有固定测量集合的整类结构化信号，恢复是成功的。然而，尽管在各种特殊情况下取得了显着的成果，但仍然缺乏对非线性观测的均匀恢复的一般理解。本文在 iid 亚高斯测量向量的假设下开发了一种统一的方法来解决这个问题。我们的主要结果表明，具有任何凸约束的简单最小二乘估计器可以用作通用恢复策略，该策略具有异常稳健性，并且不需要明确了解潜在的非线性。基于经验过程理论，一个关键的技术创新是一个近似增量条件，可以用于所有常见类型的非线性模型。这种灵活性使我们能够将我们的方法应用于非线性压缩感知和高维统计中的各种问题，从而产生一些新的和改进的保证。这些应用程序中的每一个都伴随着一个概念上简单和系统的证明，它不依赖于观察模型的任何更深层次的属性。另一方面，已知的局部稳定性属性可以以即插即用的方式整合到我们的框架中，从而暗示接近最优的误差范围。这种灵活性使我们能够将我们的方法应用于非线性压缩感知和高维统计中的各种问题，从而产生一些新的和改进的保证。这些应用程序中的每一个都伴随着一个概念上简单和系统的证明，它不依赖于观察模型的任何更深层次的属性。另一方面，已知的局部稳定性属性可以以即插即用的方式整合到我们的框架中，从而暗示接近最优的误差范围。这种灵活性使我们能够将我们的方法应用于非线性压缩感知和高维统计中的各种问题，从而产生一些新的和改进的保证。这些应用程序中的每一个都伴随着一个概念上简单和系统的证明，它不依赖于观察模型的任何更深层次的属性。另一方面，已知的局部稳定性属性可以以即插即用的方式整合到我们的框架中，从而暗示接近最优的误差范围。