In generative compressed sensing (GCS), we want to recover a signal $\mathbf{x}^* \in \mathbb{R}^n$ from $m$ measurements ($m\ll n$) using a generative prior $\mathbf{x}^*\in G(\mathbb{B}_2^k(r))$, where $G$ is typically an $L$-Lipschitz continuous generative model and $\mathbb{B}_2^k(r)$ represents the radius-$r$ $\ell_2$-ball in $\mathbb{R}^k$. Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $\mathbf{x}^*$ rather than for all $\mathbf{x}^*$ simultaneously. In this paper, we build a unified framework to derive uniform recovery guarantees for nonlinear GCS where the observation model is nonlinear and possibly discontinuous or unknown. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. Specifically, using a single realization of the sensing ensemble and generalized Lasso, {\em all} $\mathbf{x}^*\in G(\mathbb{B}_2^k(r))$ can be recovered up to an $\ell_2$-error at most $\epsilon$ using roughly $\tilde{O}({k}/{\epsilon^2})$ samples, with omitted logarithmic factors typically being dominated by $\log L$. Notably, this almost coincides with existing non-uniform guarantees up to logarithmic factors, hence the uniformity costs very little. As part of our technical contributions, we introduce the Lipschitz approximation to handle discontinuous observation models. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy. Experimental results are presented to corroborate our theory.

翻译：在生成压缩感知中，我们旨在利用生成先验从m次测量（m≪n）中恢复信号x*∈ℝ^n，其中x*∈G(𝔹₂ᵏ(r))，G通常为L-利普西茨连续生成模型，𝔹₂ᵏ(r)表示ℝᵏ中半径为r的ℓ₂球。在非线性测量条件下，现有结果多为非均匀的（即对固定x*而非全体x*同时以高概率成立）。本文构建统一框架，为观测模型非线性、甚至可能不连续或未知的非线性生成压缩感知导出均匀恢复保证。该框架以1比特/均匀量化观测及单指标模型为标准示例，具体而言：利用单次实现的感知矩阵与广义Lasso，采用约Õ(k/ε²)个样本即可使所有x*∈G(𝔹₂ᵏ(r))的ℓ₂误差不超过ε，其中省略的对数因子通常由logL主导。值得注意的是，该结果与现有非均匀保证在仅差对数因子的意义上几乎一致，表明均匀性的代价极小。作为技术贡献，我们引入利普西茨近似处理不连续观测模型，并发展出一种针对指标集具有低度量熵的乘积过程生成更紧界的集中不等式。最后给出实验验证理论结果。