This work concerns elementwise-transformations of spiked matrices: $Y_n = n^{-1/2} f(n^{1-1/(2\ell_*)} X_n + Z_n)$. Here, $f$ is a function applied elementwise, $X_n$ is a low-rank signal matrix, $Z_n$ is white noise, and $\ell_* \geq 1$ is an integer. We find that principal component analysis is powerful for recovering low-rank signal under highly non-linear and discontinuous transformations. Specifically, in the high-dimensional setting where $Y_n$ is of size $n \times p$ with $n,p \rightarrow \infty$ and $p/n \rightarrow \gamma \in (0, \infty)$, 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 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 $\gamma$. Analogous phenomena occur with $X_n$ square and symmetric and $Z_n$ a generalized Wigner matrix.

翻译：本文研究尖峰矩阵经元素级变换后的性质：$Y_n = n^{-1/2} f(n^{1-1/(2\ell_*)} X_n + Z_n)$。其中$f$为逐元素作用函数，$X_n$为低秩信号矩阵，$Z_n$为白噪声，且$\ell_* \geq 1$为整数。我们发现，在高度非线性和不连续变换下，主成分分析对低秩信号恢复具有显著有效性。具体而言，在高维设定中（$Y_n$尺寸为$n \times p$，满足$n,p \rightarrow \infty$且$p/n \rightarrow \gamma \in (0, \infty)$），我们揭示了相变现象：当信噪比超过一个依赖于$f$、$Z_n$元素分布及极限纵横比$\gamma$的尖锐阈值时，$Y_n$的主成分能够（部分）恢复$X_n$的主成分；当信噪比低于该阈值时，主成分与信号渐近正交。相比之下，直接观测$X_n + n^{-1/2}Z_n$的标准设定中，类似相变仅依赖于$\gamma$。当$X_n$为对称方阵且$Z_n$为广义Wigner矩阵时，可观察到类似现象。