New Asymptotic Results in Principal Component Analysis

Let X be a mean zero Gaussian random vector in a separable Hilbert space ℍ with covariance operator Σ : = E(X⊗ X). Let Σ = ∑ _{r ≥ 1}μ_rP_r be the spectral decomposition of Σ with distinct eigenvalues μ1>μ2>… and the corresponding spectral projectors P₁, P₂, …. Given a sample X₁, … , X_n of size n of i.i.d. copies of X, the sample covariance operator is defined as Σ̂n:=n−1∑j=1nXj⊗Xj. The main goal of principal component analysis is to estimate spectral projectors P1,P2,… by their empirical counterparts P̂1,P̂2,… properly defined in terms of spectral decomposition of the sample covariance operator Σ ̂ _n. The aim of this paper is to study asymptotic distributions of important statistics related to this problem, in particular, of statistic ∥P̂r−Pr∥22, where 22 is the squared Hilbert–Schmidt norm. This is done in a “high-complexity” asymptotic framework in which the so called effective rank r(Σ):=tr(Σ) (tr(⋅) being the trace and ∥ ⋅ ∥ _∞ being the operator norm) of the true covariance Σ is becoming large simultaneously with the sample size n, but r(Σ) = o(n) as n→ ∞. In this setting, we prove that, in the case of one-dimensional spectral projector P_r, the properly centered and normalized statistic ∥P̂r−Pr∥22 with data-dependent centering and normalization converges in distribution to a Cauchy type limit. The proofs of this and other related results rely on perturbation analysis and Gaussian concentration.