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

In Principal Component Analysis (PCA), we are given as input a matrix $A\in {\mathbb{R}}^{n\times d}$ and an integer $k\le d$, 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 $(ϵ,\delta )$-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.