Resolvent convergence for sample covariance matrices with general covariance profiles and quadratic-form control
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913120433209344 |
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| author | Louart, Cosme |
| author_facet | Louart, Cosme |
| contents | We study the resolvent \[ G^z = \left(\frac{1}{n}XX^T - zI_p\right)^{-1}, \qquad z\in\mathbb C,\ \Im(z)>0, \] where $X=(x_1,\ldots,x_n)\in\mathcal M_{p,n}$ is a random matrix with independent, but not necessarily identically distributed, columns. Our bounds are expressed in terms of moments of the centered quadratic forms \[ q_i(A):=x_i^TAx_i-\mathbb E[x_i^TAx_i], \] for deterministic matrices $A$ with unit Hilbert--Schmidt norm. In particular, we do not assume independence between the entries of a given column $x_i$. In the quasi-asymptotic regime $p\le O(n)$, the matrix $G^z$ admits a natural deterministic equivalent $\tilde G^z$, depending only on the second moments of the column vectors $x_1,\ldots,x_n$. We show that, for any deterministic matrix $B\in\mathcal M_p$, the trace $\tr(BG^z)$ is close to $\tr(B\tilde G^z)$, with error controlled by $\|B\|_{\hs}$ under first-moment bounds on the quadratic forms, and by $\|B\|_{\hs}/\sqrt n$ under suitable second-moment bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_02644 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Resolvent convergence for sample covariance matrices with general covariance profiles and quadratic-form control Louart, Cosme Probability Machine Learning 15A52, 60B12, 62J10 We study the resolvent \[ G^z = \left(\frac{1}{n}XX^T - zI_p\right)^{-1}, \qquad z\in\mathbb C,\ \Im(z)>0, \] where $X=(x_1,\ldots,x_n)\in\mathcal M_{p,n}$ is a random matrix with independent, but not necessarily identically distributed, columns. Our bounds are expressed in terms of moments of the centered quadratic forms \[ q_i(A):=x_i^TAx_i-\mathbb E[x_i^TAx_i], \] for deterministic matrices $A$ with unit Hilbert--Schmidt norm. In particular, we do not assume independence between the entries of a given column $x_i$. In the quasi-asymptotic regime $p\le O(n)$, the matrix $G^z$ admits a natural deterministic equivalent $\tilde G^z$, depending only on the second moments of the column vectors $x_1,\ldots,x_n$. We show that, for any deterministic matrix $B\in\mathcal M_p$, the trace $\tr(BG^z)$ is close to $\tr(B\tilde G^z)$, with error controlled by $\|B\|_{\hs}$ under first-moment bounds on the quadratic forms, and by $\|B\|_{\hs}/\sqrt n$ under suitable second-moment bounds. |
| title | Resolvent convergence for sample covariance matrices with general covariance profiles and quadratic-form control |
| topic | Probability Machine Learning 15A52, 60B12, 62J10 |
| url | https://arxiv.org/abs/2109.02644 |