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In generative compressed sensing (GCS), we want to recover a signal$\mathbf{x}^* \in \mathbb{R}^n$ from $m$ measurements ($m\ll n$) using agenerative prior $\mathbf{x}^*\in G(\mathbb{B}_2^k(r))$, where $G$ is typicallyan $L$-Lipschitz continuous generative model and $\mathbb{B}_2^k(r)$ representsthe 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 afixed $\mathbf{x}^*$ rather than for all $\mathbf{x}^*$ simultaneously. In thispaper, we build a unified framework to derive uniform recovery guarantees fornonlinear GCS where the observation model is nonlinear and possiblydiscontinuous or unknown. Our framework accommodates GCS with 1-bit/uniformlyquantized observations and single index models as canonical examples.Specifically, using a single realization of the sensing ensemble andgeneralized Lasso, {\em all} $\mathbf{x}^*\in G(\mathbb{B}_2^k(r))$ can berecovered up to an $\ell_2$-error at most $\epsilon$ using roughly$\tilde{O}({k}/{\epsilon^2})$ samples, with omitted logarithmic factorstypically being dominated by $\log L$. Notably, this almost coincides withexisting non-uniform guarantees up to logarithmic factors, hence the uniformitycosts very little. As part of our technical contributions, we introduce theLipschitz approximation to handle discontinuous observation models. We alsodevelop a concentration inequality that produces tighter bounds for productprocesses whose index sets have low metric entropy. Experimental results arepresented to corroborate our theory.

A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing