New asymptotic results in principal component analysis

الأصل · EN

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≥ ₁μᵣ Pᵣ be the spectral decomposition of Σ with distinct eigenvalues μ₁>μ₂> and the corresponding spectral projectors P₁, P₂,. Given a sample X₁,, Xₙ of size n of i.i.d. copies of X, the sample covariance operator is defined as Σₙ:= n⁻¹∑ⱼ₌₁ⁿ Xⱼ⊗ Xⱼ. The main goal of principal component analysis is to estimate spectral projectors P₁, P₂, by their empirical counterparts P₁, P₂, properly defined in terms of spectral decomposition of the sample covariance operator Σₙ. The aim of this paper is to study asymptotic distributions of important statistics related to this problem, in particular, of statistic Pᵣ-Pᵣ₂², 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)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 Pᵣ, the properly centered and normalized statistic Pᵣ-Pᵣ₂² 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.