  This work concerns elementwise-transformations of spiked matrices: $Y_n =
n^{-1/2} f( \sqrt{n} X_n + Z_n)$. Here, $f$ is a function applied elementwise,
$X_n$ is a low-rank signal matrix, and $Z_n$ is white noise. We find that
principal component analysis is powerful for recovering signal under highly
nonlinear or discontinuous transformations. Specifically, in the
high-dimensional setting where $Y_n$ is of size $n \times p$ with $n,p
\rightarrow \infty$ and $p/n \rightarrow \gamma > 0$, we uncover a phase
transition: for signal-to-noise ratios above a sharp threshold -- depending on
$f$, the distribution of elements of $Z_n$, and the limiting aspect ratio
$\gamma$ -- the principal components of $Y_n$ (partially) recover those of
$X_n$. Below this threshold, the principal components of $Y_n$ are
asymptotically orthogonal to the signal. In contrast, in the standard setting
where $X_n + n^{-1/2}Z_n$ is observed directly, the analogous phase transition
depends only on $\gamma$. A similar phenomenon occurs with $X_n$ square and
symmetric and $Z_n$ a generalized Wigner matrix.
