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Differentially Private Preconditioned Power Iteration Method for Principal Component Analysis that Adapts to the Coherence

Duy V. Le, Ta Duy Nguyen, Alina Ene, Huy L. Nguyen

In Principal Component Analysis (PCA), we are given as input a matrix $A\in\mathbb{R}^{n\times d}$ and an integer $k\leq d$, and the goal is to find the top $k$ right-singular vectors of $A$. We consider the setting where each row of $A$ is the private data of an individual, and we design an $(ε,δ)$-differentially private algorithm whose utility guarantee improves with the coherence of the input matrix. Prior works were either restricted to the case $k=1$ or ensured significantly weaker privacy protections. Our algorithm is based on a preconditioned variant of the power iteration method that is designed to significantly reduce the noise needed to ensure privacy.

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