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.

翻译：本文关注尖峰矩阵的逐元素变换：$Y_n = n^{-1/2} f( \sqrt{n} X_n + Z_n)$。其中，$f$为逐元素施加的函数，$X_n$为低秩信号矩阵，$Z_n$为白噪声。研究发现，在高度非线性或不连续变换下，主成分分析对信号恢复具有显著效能。具体而言，在高维情形下（$Y_n$规模为$n \times p$，且$n,p \rightarrow \infty$，$p/n \rightarrow \gamma > 0$），我们揭示了一个相变现象：当信噪比超过一个尖锐阈值（该阈值取决于$f$、$Z_n$元素的分布以及极限纵横比$\gamma$）时，$Y_n$的主成分能（部分）恢复$X_n$的主成分；低于该阈值时，$Y_n$的主成分与信号渐近正交。与之对比，在直接观测$X_n + n^{-1/2}Z_n$的标准设定中，类似的相变仅由$\gamma$决定。当$X_n$为方对称矩阵且$Z_n$为广义Wigner矩阵时，亦存在类似现象。