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Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models. (arXiv:1807.00891v1 [math.ST])
来源于:arXiv
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
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