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• W2323628131 abstract "We compute analytically, for large $N$, the probability $mathcal{P}({N}_{+},N)$ that a $Nifmmodetimeselsetexttimesfi{}N$ Wishart random matrix has ${N}_{+}$ eigenvalues exceeding a threshold $Nensuremath{zeta}$, including its large deviation tails. This probability plays a benchmark role when performing the principal component analysis of a large empirical data set. We find that $mathcal{P}({N}_{+},N)ensuremath{approx}mathrm{exp}[ensuremath{-}ensuremath{beta}{N}^{2}{ensuremath{psi}}_{ensuremath{zeta}}({N}_{+}/N)]$, where $ensuremath{beta}$ is the Dyson index of the ensemble and ${ensuremath{psi}}_{ensuremath{zeta}}(ensuremath{kappa})$ is a rate function that we compute explicitly in the full range $0ensuremath{le}ensuremath{kappa}ensuremath{le}1$ and for any $ensuremath{zeta}$. The rate function ${ensuremath{psi}}_{ensuremath{zeta}}(ensuremath{kappa})$ displays a quadratic behavior modulated by a logarithmic singularity close to its minimum ${ensuremath{kappa}}^{ensuremath{star}}(ensuremath{zeta})$. This is shown to be a consequence of a phase transition in an associated Coulomb gas problem. The variance $ensuremath{Delta}(N)$ of the number of relevant components is also shown to grow universally (independent of $ensuremath{zeta}$) as $ensuremath{Delta}(N)ensuremath{sim}(ensuremath{beta}{ensuremath{pi}}^{2}{)}^{ensuremath{-}1}mathrm{ln}N$ for large $N$." @default.
• W2323628131 created "2016-06-24" @default.
• W2323628131 creator A5029474578 @default.
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• W2323628131 date "2012-05-18" @default.
• W2323628131 modified "2023-10-18" @default.
• W2323628131 title "Number of Relevant Directions in Principal Component Analysis and Wishart Random Matrices" @default.
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• W2323628131 doi "https://doi.org/10.1103/physrevlett.108.200601" @default.
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