On Purely Private Covariance Estimation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908804154654720 |
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| author | d'Orsi, Tommaso Novikov, Gleb |
| author_facet | d'Orsi, Tommaso Novikov, Gleb |
| contents | We present a simple perturbation mechanism for the release of $d$-dimensional covariance matrices $Σ$ under pure differential privacy. For large datasets with at least $n\geq d^2/\varepsilon$ elements, our mechanism recovers the provably optimal Frobenius norm error guarantees of \cite{nikolov2023private}, while simultaneously achieving best known error for all other $p$-Schatten norms, with $p\in [1,\infty]$. Our error is information-theoretically optimal for all $p\ge 2$, in particular, our mechanism is the first purely private covariance estimator that achieves optimal error in spectral norm.
For small datasets $n< d^2/\varepsilon$, we further show that by projecting the output onto the nuclear norm ball of appropriate radius, our algorithm achieves the optimal Frobenius norm error $O(\sqrt{d\;\text{Tr}(Σ) /n})$, improving over the known bounds of $O(\sqrt{d/n})$ of \cite{nikolov2023private} and ${O}\big(d^{3/4}\sqrt{\text{Tr}(Σ)/n}\big)$ of \cite{dong2022differentially}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26717 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Purely Private Covariance Estimation d'Orsi, Tommaso Novikov, Gleb Machine Learning Data Structures and Algorithms We present a simple perturbation mechanism for the release of $d$-dimensional covariance matrices $Σ$ under pure differential privacy. For large datasets with at least $n\geq d^2/\varepsilon$ elements, our mechanism recovers the provably optimal Frobenius norm error guarantees of \cite{nikolov2023private}, while simultaneously achieving best known error for all other $p$-Schatten norms, with $p\in [1,\infty]$. Our error is information-theoretically optimal for all $p\ge 2$, in particular, our mechanism is the first purely private covariance estimator that achieves optimal error in spectral norm. For small datasets $n< d^2/\varepsilon$, we further show that by projecting the output onto the nuclear norm ball of appropriate radius, our algorithm achieves the optimal Frobenius norm error $O(\sqrt{d\;\text{Tr}(Σ) /n})$, improving over the known bounds of $O(\sqrt{d/n})$ of \cite{nikolov2023private} and ${O}\big(d^{3/4}\sqrt{\text{Tr}(Σ)/n}\big)$ of \cite{dong2022differentially}. |
| title | On Purely Private Covariance Estimation |
| topic | Machine Learning Data Structures and Algorithms |
| url | https://arxiv.org/abs/2510.26717 |