Efficient estimation of linear functionals of principal components

We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_{1},\dots,X_{n}$ in a separable Hilbert space $\mathbb{H}$ with unknown covariance operator $\Sigma$ . The complexity of the problem is characterized by its effective rank $\mathbf{r}(\Sigma):=\frac{\operatorname{tr}(\Sigma)}{\|\Sigma \|}$ , where $\mathrm{tr}(\Sigma)$ denotes the trace of $\Sigma$ and $\|\Sigma\|$ denotes its operator norm. We develop a method of bias reduction in the problem of estimation of linear functionals of eigenvectors of $\Sigma$ . Under the assumption that $\mathbf{r}(\Sigma)=o(n)$ , we establish the asymptotic normality and asymptotic properties of the risk of the resulting estimators and prove matching minimax lower bounds, showing their semiparametric optimality.We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_{1},\dots,X_{n}$ in a separable Hilbert space $\mathbb{H}$ with unknown covariance operator $\Sigma $. The complexity of the problem is characterized by its effective rank $\mathbf{r}(\Sigma):=\frac{\operatorname{tr}(\Sigma)}{\|\Sigma \|}$, where $\mathrm{tr}(\Sigma)$ denotes the trace of $\Sigma $ and $\|\Sigma\|$ denotes its operator norm. We develop a method of bias reduction in the problem of estimation of linear functionals of eigenvectors of $\Sigma $. Under the assumption that $\mathbf{r}(\Sigma)=o(n)$, we establish the asymptotic normality and asymptotic properties of the risk of the resulting estimators and prove matching minimax lower bounds, showing their semiparametric optimality.

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笔记与分组

分类依据与纠错

Efficient estimation of linear functionals of principal components

代码、数据与补充材料

主成分线性泛函的有效估计

原文摘要

同方向的基础文献

同方向的基础文献