Number of Relevant Directions in Principal Component Analysis and Wishart Random Matrices
Number of Relevant Directions in Principal Component Analysis and Wishart Random Matrices
We compute analytically, for large N, the probability P(N+,N) that a N×N Wishart random matrix has N+ eigenvalues exceeding a threshold Nζ, 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 P(N+,N)≈exp[-βN2ψζ(N+/N)], where …