A central problem of random matrix theory is to understand the eigenvalues of
spiked random matrix models, introduced by Johnstone, in which a prominent
eigenvector (or "spike") is planted into a random matrix. These distributions
form natural statistical models for principal component analysis (PCA) problems
throughout the sciences. Baik, Ben Arous and Peche showed that the spiked
Wishart ensemble exhibits a sharp phase transition asymptotically: when the
spike strength is above a critical threshold, it is possible to detect the
presence of a spike based on the top eigenvalue, and below the threshold the
top eigenvalue provides no information. Such results form the basis of our
understanding of when PCA can detect a low-rank signal in the presence of
noise. However, under structural assumptions on the spike, not all information
is necessarily contained in the spectrum. We study the statistical limits of
tests for the presence of a spike, including non-spectral tests. Our results
leverage Le Cam's notion of contiguity, and include:
i) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal
detection threshold for certain natural priors for the spike.
ii) For any non-Gaussian Wigner ensemble, PCA is sub-optimal for detection.
However, an efficient variant of PCA achieves the optimal threshold (for
natural priors) by pre-transforming the matrix entries.
iii) For the Gaussian Wishart ensemble, the PCA threshold is optimal for
positive spikes (for natural priors) but this is not always the case for
negative spikes.

通过研究 Le Cam 的连续性，我们探讨了用于检测稀疏向量存在的测试的统计极限，包括非光谱测试，并证明了对于高斯 Wishart 集合，PCA 阈值对于正向 spikes（自然先验）是最佳的，但这并不总是负向 spikes 的情况。