Eigenvalue distribution of the Hadamard product of sample covariance matrices in a quadratic regime
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912235829329920 |
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| author | Assaly, Sebastien Abou Benigni, Lucas |
| author_facet | Assaly, Sebastien Abou Benigni, Lucas |
| contents | In this note, we prove that if $X\in\mathbb{R}^{n\times d}$ and $Y\in\mathbb{R}^{n\times p}$ are two independent matrices with i.i.d entries then the empirical spectral distribution of $\frac{1}{d}XX^\top \odot \frac{1}{p}YY^\top$, where $\odot$ denotes the Hadamard product, converges to the Marchenko--Pastur distribution of shape $γ$ in the quadratic regime of dimension $\frac{n}{dp}\to γ$ and $\frac{p}{d}\to a$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12374 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenvalue distribution of the Hadamard product of sample covariance matrices in a quadratic regime Assaly, Sebastien Abou Benigni, Lucas Probability 60B20, 15B52 In this note, we prove that if $X\in\mathbb{R}^{n\times d}$ and $Y\in\mathbb{R}^{n\times p}$ are two independent matrices with i.i.d entries then the empirical spectral distribution of $\frac{1}{d}XX^\top \odot \frac{1}{p}YY^\top$, where $\odot$ denotes the Hadamard product, converges to the Marchenko--Pastur distribution of shape $γ$ in the quadratic regime of dimension $\frac{n}{dp}\to γ$ and $\frac{p}{d}\to a$. |
| title | Eigenvalue distribution of the Hadamard product of sample covariance matrices in a quadratic regime |
| topic | Probability 60B20, 15B52 |
| url | https://arxiv.org/abs/2502.12374 |