Privacy via the Johnson-Lindenstrauss Transform

Abstract

Suppose that party A collects private information about its users, where each user's data is represented as a bit vector. Suppose that party B has a proprietary data mining algorithm that requires estimating the distance between users, such as clustering or nearest neighbors. We ask if it is possible for party A to publish some information about each user so that B can estimate the distance between users without being able to infer any private bit of a user. Our method involves projecting each user's representation into a random, lower-dimensional space via a sparse Johnson-Lindenstrauss transform and then adding Gaussian noise to each entry of the lower-dimensional representation. We show that the method preserves differential privacy-where the more privacy is desired, the larger the variance of the Gaussian noise. Further, we show how to approximate the true distances between users via only the lower-dimensional, perturbed data. Finally, we consider other perturbation methods such as randomized response and draw comparisons to sketch-based methods. While the goal of releasing user-specific data to third parties is more broad than preserving distances, this work shows that distance computations with privacy is an achievable goal.

Challenging Differential Privacy:The Case of Non-interactive Mechanisms

“…Kenthapadi and co-authors [2] proposed another mechanism to prevent the adversary from learning the presence (or absence) of an item in the profile, although their scheme tackles a different data type (i.e., real vector). In the sequel, we denote this proposal by JLT because it is based on the Johnson-Lindenstrauss Transform.…”

“…The first mechanism is based on randomizing a Bloom filter representation of the profile [1] while the second relies on the application of the Johnson-Lindenstrauss transform and the addition of noise [2]. Both mechanisms preserve some global properties such as the ability to compute a distance between two profiles while hiding the details of the profiles themselves.…”

“…Profile P is encoded into vector Y P = j∈P X j , then the user adds a noise N (private data) before publishingỸ P = Y P + N. The authors of [2] recommend a white Gaussian noise:…”

“…First in Section 2, we give an overview of the concept of differential privacy. Then, we describe two noninteractive differentially private mechanisms in Sections 3 and 4 that have been recently proposed: BLIP (BLoom-and-flIP ) [1] and one based on the JohnsonLindenstrauss Transform (JLT in short) [2]. These mechanisms transform the profile of a user into a compact representation that estimate the similarity between profiles while hiding the presence or absence of a particular item in the profile (in the sense of differential privacy).…”

“…One possible approach to solve this problem is to sanitize the profile of a user with a non-interactive mechanism compliant with the concept of differential privacy before his publication. In particular, this paper investigates the privacy guarantees offered by two non-interactive mechanisms offering a differentially private representation of profiles: BLIP (BLoom-and-flIP) [1] and JLT (Johnson-Lindenstrauss Transform) [2].…”