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A simple population eigengap gives root-n sample PCA eigenspace consistency

Statement

Let $X_{n, 1}, \dots, X_{n, n}$ be iid copies of a fixed $X \in R^{p}$ with $E ∥ X ∥_{2}^{4} < \infty$ , where $p \geq 2$ , and let $Σ = Cov (X)$ have a simple leading eigenvalue with gap $g = λ_{1} (Σ) - λ_{2} (Σ) > 0$ . Let ${\hat{Σ}}_{n}^{*}$ be the sample-centered covariance with divisor $n$ , and put $ϵ_{n} = ∥ {\hat{Σ}}_{n}^{*} - Σ ∥_{o p}$ and $G_{n} = {ϵ_{n} < g / 2}$ . Then $P (G_{n}) \to 1$ . On $G_{n}$ the sample leading eigenspace is a line; define $Z_{n}$ to be the sine of its angle to the population leading line and set $Z_{n} = 0$ off $G_{n}$ . Then $Z_{n} = O_{P} (n^{- 1 / 2})$ , hence $Z_{n} \to 0$ in probability. In particular, the sample leading eigenspace is a unique line with probability tending to one. The angle is line-based and sign-invariant. This is a local eigenspace-consistency result, not finite-sample inference for a selected component.

Facts & Assumptions

Given: Fixed $p \geq 2$ , iid samples from a law with finite fourth moment, the population covariance $Σ$ , its simple leading eigenvalue gap $g > 0$ , and sample-centered covariance ${\hat{Σ}}_{n}^{*}$ with divisor $n$ .

[F1]

Finite-fourth-moment root-n consistency of sample-centered covariance: $∥ {\hat{Σ}}_{n}^{*} - Σ ∥_{F} = O_{P} (n^{- 1 / 2})$ .

[F2]

Local leading-eigenvector stability under a spectral gap: if a symmetric perturbation has operator norm $ϵ < g / 2$ , the perturbed leading eigenvalue remains simple and its leading line has sine-angle at most $ϵ / (g - ϵ)$ from the population line.

[F3]

Sample-centered covariance is a measurable statistic and estimator: the sample covariance is a measurable matrix-valued statistic.

[F4]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: $∥ B ∥_{o p} = \max_{∥ u ∥_{2} = 1} ∥ B u ∥_{2}$ .

[F5]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: the Euclidean norm is induced by the inner product, and Cauchy--Schwarz holds.

[F6]

Stochastic order and convergence notation for finite-dimensional random vectors: $O_{P} (1)$ means bounded in probability; thus an $O_{P} (n^{- 1 / 2})$ quantity becomes $O_{P} (1)$ after multiplication by $\sqrt{n}$ .

Proof

technique · direct

1.1F1F3F4F5F6givenalgebra

For every $p \times p$ matrix $B$ and unit vector $u$ , applying Cauchy--Schwarz to each row gives $∥ B u ∥_{2}^{2} = \sum_{j} (\sum_{k} B_{j k} u_{k})^{2} \leq \sum_{j, k} B_{j k}^{2} = ∥ B ∥_{F}^{2}$ . Hence [F4] gives $∥ B ∥_{o p} \leq ∥ B ∥_{F}$ . With $ϵ_{n} = ∥ {\hat{Σ}}_{n}^{*} - Σ ∥_{o p}$ , [F1] implies $ϵ_{n} = O_{P} (n^{- 1 / 2})$ by [F6]. To see $P (G_{n}) \to 1$ , fix $η > 0$ and choose $C$ so the limsup tail of $\sqrt{n} ∥ {\hat{Σ}}_{n}^{*} - Σ ∥_{F}$ above $C$ is at most $η$ . For all sufficiently large $n$ , $C / \sqrt{n} < g / 2$ , and then $G_{n}^{c} \subseteq {\sqrt{n} ∥ {\hat{Σ}}_{n}^{*} - Σ ∥_{F} > C}$ . Thus ${lim sup}_{n} P (G_{n}^{c}) \leq η$ , so the claimed probability tends to one.

2.1F1F2F3F6step 1.1algebra

On $G_{n}$ , [F2] says that the sample leading eigenvalue is simple and $Z_{n} \leq ϵ_{n} / (g - ϵ_{n}) \leq 2 ϵ_{n} / g$ . The sample matrix is measurable by [F3]; its operator-norm distance from $Σ$ is therefore measurable. The leading line varies continuously on the set of matrices with a simple top eigenvalue: applying [F2] with any such matrix as the base and a perturbation tending to zero makes the sign-free line angle tend to zero. Hence $Z_{n}$ , extended by zero off the Borel event $G_{n}$ , is measurable. Pointwise, $\sqrt{n} Z_{n} \leq (2 / g) \sqrt{n} ϵ_{n}$ , so [F1] and [F6] imply $\sqrt{n} Z_{n} = O_{P} (1)$ and therefore $Z_{n} = O_{P} (n^{- 1 / 2})$ .

3.1F6step 2.1algebra∎

Fix $δ > 0$ and $η > 0$ . Since [F6] defines $\sqrt{n} Z_{n} = O_{P} (1)$ , choose $C$ so its limsup tail above $C$ is at most $η$ . For all sufficiently large $n$ , $C / \sqrt{n} < δ$ , and then ${Z_{n} > δ} \subseteq {\sqrt{n} Z_{n} > C}$ . Hence ${lim sup}_{n} P (Z_{n} > δ) \leq η$ . Since $η$ was arbitrary, $Z_{n} \to 0$ in probability.

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Sources

Online Bayesian Data Analysis, Lecture Notes 8: Learning Signal Representations (11 April 2016) (standard reference, not scraped)
Soumendu Sundar Mukherjee, Multivariate Analysis, Lecture 8: Principal Component Analysis (standard reference, not scraped)