Number of relevant directions in Principal Component Analysis and Wishart random matrices

Abstract

We compute analytically, for large $N$, the probability $\mathcal{P}(N_{+},N)$ that a $N\times N$ Wishart random matrix has $N_{+}$ eigenvalues exceeding a threshold $N\zeta$, including its large deviation tails. This probability plays a benchmark role when performing the Principal Component Analysis of a large empirical dataset. We find that $\mathcal{P}(N_{+},N)\approx \exp (-\beta N^2{\psi }_{\zeta }(N_{+}/N))$, where $\beta$ is the Dyson index of the ensemble and ${\psi }_{\zeta }(\kappa )$ is a rate function that we compute explicitly in the full range $0\le \kappa \le 1$ and for any $\zeta$. The rate function ${\psi }_{\zeta }(\kappa )$ displays a quadratic behavior modulated by a logarithmic singularity close to its minimum ${\kappa }^{\star }(\zeta )$. This is shown to be a consequence of a phase transition in an associated Coulomb gas problem. The variance $\Delta (N)$ of the number of relevant components is also shown to grow universally (independent of $\zeta )$ as $\Delta (N)\sim (\beta {\pi }^2{)}^{-1}\ln N$ for large $N$.

5 pag., 2 fig