  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.
