Private Covariance Approximation and Eigenvalue-Gap Bounds for Complex Gaussian Perturbations

Oren Mangoubi, Nisheeth K. Vishnoi
Proceedings of Thirty Sixth Conference on Learning Theory, PMLR 195:1522-1587, 2023.

Abstract

We consider the problem of approximating a $d \times d$ covariance matrix $M$ with a rank-$k$ matrix under $(\varepsilon,\delta)$-differential privacy. We present and analyze a complex variant of the Gaussian mechanism and show that the Frobenius norm of the difference between the matrix output by this mechanism and the best rank-$k$ approximation to $M$ is bounded by roughly $\tilde{O}(\sqrt{kd})$, whenever there is an appropriately large gap between the $k$’th and the $k+1$’th eigenvalues of $M$. This improves on previous work that requires that the gap between every pair of top-$k$ eigenvalues of $M$ is at least $\sqrt{d}$ for a similar bound. Our analysis leverages the fact that the eigenvalues of complex matrix Brownian motion repel more than in the real case, and uses Dyson’s stochastic differential equations governing the evolution of its eigenvalues to show that the eigenvalues of the matrix $M$ perturbed by complex Gaussian noise have large gaps with high probability. Our results contribute to the analysis of low-rank approximations under average-case perturbations and to an understanding of eigenvalue gaps for random matrices, which may be of independent interest.

Cite this Paper

BibTeX
@InProceedings{pmlr-v195-mangoubi23a, title = {Private Covariance Approximation and Eigenvalue-Gap Bounds for Complex Gaussian Perturbations}, author = {Mangoubi, Oren and Vishnoi, Nisheeth K.}, booktitle = {Proceedings of Thirty Sixth Conference on Learning Theory}, pages = {1522--1587}, year = {2023}, editor = {Neu, Gergely and Rosasco, Lorenzo}, volume = {195}, series = {Proceedings of Machine Learning Research}, month = {12--15 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v195/mangoubi23a/mangoubi23a.pdf}, url = {https://proceedings.mlr.press/v195/mangoubi23a.html}, abstract = {We consider the problem of approximating a $d \times d$ covariance matrix $M$ with a rank-$k$ matrix under $(\varepsilon,\delta)$-differential privacy. We present and analyze a complex variant of the Gaussian mechanism and show that the Frobenius norm of the difference between the matrix output by this mechanism and the best rank-$k$ approximation to $M$ is bounded by roughly $\tilde{O}(\sqrt{kd})$, whenever there is an appropriately large gap between the $k$’th and the $k+1$’th eigenvalues of $M$. This improves on previous work that requires that the gap between every pair of top-$k$ eigenvalues of $M$ is at least $\sqrt{d}$ for a similar bound. Our analysis leverages the fact that the eigenvalues of complex matrix Brownian motion repel more than in the real case, and uses Dyson’s stochastic differential equations governing the evolution of its eigenvalues to show that the eigenvalues of the matrix $M$ perturbed by complex Gaussian noise have large gaps with high probability. Our results contribute to the analysis of low-rank approximations under average-case perturbations and to an understanding of eigenvalue gaps for random matrices, which may be of independent interest.} }
Endnote
%0 Conference Paper %T Private Covariance Approximation and Eigenvalue-Gap Bounds for Complex Gaussian Perturbations %A Oren Mangoubi %A Nisheeth K. Vishnoi %B Proceedings of Thirty Sixth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2023 %E Gergely Neu %E Lorenzo Rosasco %F pmlr-v195-mangoubi23a %I PMLR %P 1522--1587 %U https://proceedings.mlr.press/v195/mangoubi23a.html %V 195 %X We consider the problem of approximating a $d \times d$ covariance matrix $M$ with a rank-$k$ matrix under $(\varepsilon,\delta)$-differential privacy. We present and analyze a complex variant of the Gaussian mechanism and show that the Frobenius norm of the difference between the matrix output by this mechanism and the best rank-$k$ approximation to $M$ is bounded by roughly $\tilde{O}(\sqrt{kd})$, whenever there is an appropriately large gap between the $k$’th and the $k+1$’th eigenvalues of $M$. This improves on previous work that requires that the gap between every pair of top-$k$ eigenvalues of $M$ is at least $\sqrt{d}$ for a similar bound. Our analysis leverages the fact that the eigenvalues of complex matrix Brownian motion repel more than in the real case, and uses Dyson’s stochastic differential equations governing the evolution of its eigenvalues to show that the eigenvalues of the matrix $M$ perturbed by complex Gaussian noise have large gaps with high probability. Our results contribute to the analysis of low-rank approximations under average-case perturbations and to an understanding of eigenvalue gaps for random matrices, which may be of independent interest.
APA
Mangoubi, O. & Vishnoi, N.K.. (2023). Private Covariance Approximation and Eigenvalue-Gap Bounds for Complex Gaussian Perturbations. Proceedings of Thirty Sixth Conference on Learning Theory, in Proceedings of Machine Learning Research 195:1522-1587 Available from https://proceedings.mlr.press/v195/mangoubi23a.html.

Related Material