While it is well known that the restricted isometry property (RIP) guarantees uniform sparse recovery from noisy linear measurements, uniform recovery of structured signals from nonlinear observations remains much less understood. This paper shows that the restricted approximate invertibility condition (RAIC) provides a unified approach to this end. Particularly, uniform recovery is achieved by projected gradient descent (PGD) with gradients obeying RAIC for all signals. As an application, under a large class of piecewise Lipschitz link functions (possibly discontinuous), we develop a uniform recovery theory for Gaussian single-index model by establishing the uniform RAIC for the gradient of the (scaled) $\ell_2$ loss via a covering argument. The theory generalizes the nonuniform recovery guarantees due to Plan and Vershynin (2016); Oymak and Soltanolkotabi (2017) and exhibits additional error terms that can be interpreted as the cost of uniform recovery. Intriguingly, in the three canonical settings of (a) sparse recovery via PGD with $\ell_0$ projection (i.e., iterative hard thresholding (IHT)), (b) sparse recovery via PGD with $\ell_1$ projection, and (c) recovering approximately sparse signals via PGD with $\ell_1$ projection, the additional error terms are negligible and in turn our uniform recovery error rates are at the same order of existing nonuniform ones, up to log factors. Our results hence improve on Genzel and Stollenwerk (2023). Under the specific nonlinearity of 1-bit quantization, we use a VC dimension argument to show that the uniform recovery error of IHT is at the same order of the nonuniform recovery error, with no loss of log factor. In addition, we show that the robustness of PGD to noise and corruption can be incorporated elegantly by bounding a single additional random process that captures the gradient mismatch.

翻译：尽管众所周知受限等距性质（RIP）可保证从含噪线性测量中均匀地恢复稀疏信号，但从非线性观测中均匀恢复结构信号的理解仍远不充分。本文证明受限近似可逆条件（RAIC）为此提供了一种统一方法。特别地，通过投影梯度下降（PGD）算法实现均匀恢复，该算法对所有信号均满足RAIC条件。作为应用，针对一大类分段Lipschitz连接函数（可能不连续），我们通过覆盖论证建立了（尺度化）$\ell_2$损失梯度的均匀RAIC，从而发展了高斯单指标模型的均匀恢复理论。该理论推广了Plan与Vershynin（2016）以及Oymak与Soltanolkotabi（2017）的非均匀恢复保证，并展现出可解释为均匀恢复代价的额外误差项。引人注目的是，在三种典型设置下：（a）采用$\ell_0$投影的PGD稀疏恢复（即迭代硬阈值（IHT）），（b）采用$\ell_1$投影的PGD稀疏恢复，以及（c）采用$\ell_1$投影的PGD近似稀疏信号恢复，这些额外误差项可忽略不计，从而使我们的均匀恢复误差率与现有非均匀恢复误差率（考虑对数因子）同阶。因此，我们的结果改进了Genzel与Stollenwerk（2023）的工作。针对1比特量化这一特定非线性情形，我们使用VC维论证表明IHT的均匀恢复误差与非均匀恢复误差同阶，且无对数因子损失。此外，我们证明通过约束单个刻画梯度失配的额外随机过程，可优雅地纳入PGD对噪声与污染的鲁棒性。