The Johnson-Lindenstrauss Transform Itself Preserves Differential Privacy

Abstract

This paper proves that an "old dog", namely -- the classical Johnson-Lindenstrauss transform, "performs new tricks" -- it gives a novel way of preserving differential privacy. We show that if we take two databases, $D$ and $D'$, such that (i) $D'-D$ is a rank-1 matrix of bounded norm and (ii) all singular values of $D$ and $D'$ are sufficiently large, then multiplying either $D$ or $D'$ with a vector of iid normal Gaussians yields two statistically close distributions in the sense of differential privacy. Furthermore, a small, deterministic and \emph{public} alteration of the input is enough to assert that all singular values of $D$ are large. We apply the Johnson-Lindenstrauss transform to the task of approximating cut-queries: the number of edges crossing a $(S,\bar S)$-cut in a graph. We show that the JL transform allows us to \emph{publish a sanitized graph} that preserves edge differential privacy (where two graphs are neighbors if they differ on a single edge) while adding only $O(|S|/\epsilon)$ random noise to any given query (w.h.p). Comparing the additive noise of our algorithm to existing algorithms for answering cut-queries in a differentially private manner, we outperform all others on small cuts ($|S| = o(n)$). We also apply our technique to the task of estimating the variance of a given matrix in any given direction. The JL transform allows us to \emph{publish a sanitized covariance matrix} that preserves differential privacy w.r.t bounded changes (each row in the matrix can change by at most a norm-1 vector) while adding random noise of magnitude independent of the size of the matrix (w.h.p). In contrast, existing algorithms introduce an error which depends on the matrix dimensions.