The paper claims an improved differentially private rank-k covariance approximation via complex Gaussian noise and Dyson Brownian motion, but its real-part output can have rank up to 2k, so the main theorem as stated is unsupported.
The Johnson-Lindenstrauss transform itself preserves differential privacy
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Private Low-Rank Approximation for Covariance Matrices, Dyson Brownian Motion, and Eigenvalue-Gap Bounds for Gaussian Perturbations
The paper claims an improved differentially private rank-k covariance approximation via complex Gaussian noise and Dyson Brownian motion, but its real-part output can have rank up to 2k, so the main theorem as stated is unsupported.