The Johnson-Lindenstrauss Transform Itself Preserves Differential Privacy

September 19, 2012

We show 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 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 topublish 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 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.

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 topublish 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 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.