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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.18173 |
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| _version_ | 1866911963103100928 |
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| author | Lin, Zeqin Pan, Guangming |
| author_facet | Lin, Zeqin Pan, Guangming |
| contents | Consider a data matrix $Y = [\mathbf{y}_1, \cdots, \mathbf{y}_N]$ of size $M \times N$, where the columns are independent observations from a random vector $\mathbf{y}$ with zero mean and population covariance $Σ$. Let $\mathbf{u}_i$ and $\mathbf{v}_j$ denote the left and right singular vectors of $Y$, respectively. This study investigates the eigenvector/singular vector overlaps $\langle {\mathbf{u}_i, D_1 \mathbf{u}_j} \rangle$, $\langle {\mathbf{v}_i, D_2 \mathbf{v}_j} \rangle$ and $\langle {\mathbf{u}_i, D_3 \mathbf{v}_j} \rangle$, where $D_k$ are general deterministic matrices with bounded operator norms. We establish the convergence in probability of these eigenvector overlaps toward their deterministic counterparts with explicit convergence rates, when the dimension $M$ scales proportionally with the sample size $N$. Building on these findings, we offer a more precise characterization of the loss for Ledoit and Wolf's nonlinear shrinkage estimators of the population covariance $Σ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18173 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eigenvector overlaps in large sample covariance matrices and nonlinear shrinkage estimators Lin, Zeqin Pan, Guangming Statistics Theory Consider a data matrix $Y = [\mathbf{y}_1, \cdots, \mathbf{y}_N]$ of size $M \times N$, where the columns are independent observations from a random vector $\mathbf{y}$ with zero mean and population covariance $Σ$. Let $\mathbf{u}_i$ and $\mathbf{v}_j$ denote the left and right singular vectors of $Y$, respectively. This study investigates the eigenvector/singular vector overlaps $\langle {\mathbf{u}_i, D_1 \mathbf{u}_j} \rangle$, $\langle {\mathbf{v}_i, D_2 \mathbf{v}_j} \rangle$ and $\langle {\mathbf{u}_i, D_3 \mathbf{v}_j} \rangle$, where $D_k$ are general deterministic matrices with bounded operator norms. We establish the convergence in probability of these eigenvector overlaps toward their deterministic counterparts with explicit convergence rates, when the dimension $M$ scales proportionally with the sample size $N$. Building on these findings, we offer a more precise characterization of the loss for Ledoit and Wolf's nonlinear shrinkage estimators of the population covariance $Σ$. |
| title | Eigenvector overlaps in large sample covariance matrices and nonlinear shrinkage estimators |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2404.18173 |