The Johnson-Lindenstrauss Transform Itself Preserves Differential Privacy

TLDR

Existing algorithms for differential privacy introduce error that scales with matrix dimensions. The paper applies the Johnson–Lindenstrauss transform to approximate graph cut queries and to estimate matrix variances. The authors prove that multiplying a rank‑1 perturbed database by a Gaussian vector yields statistically close distributions, and they use the JL transform—after a deterministic public tweak that guarantees large singular values—to produce sanitized outputs. They show that the JL transform preserves differential privacy, enabling a sanitized graph with only O(|S|ϵ) noise for cut queries (outperforming prior methods on small cuts) and a sanitized covariance matrix with noise independent of matrix size.

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 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, S)-cut in a graph. We show that the JL transform allows us to 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|ϵ) 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.