TY - JOUR

T1 - Representing Sparse Vectors with Differential Privacy, Low Error, Optimal Space, and Fast Access

AU - Aumüller, Martin

AU - Lebeda, Christian Janos

AU - Pagh, Rasmus

N1 - Publisher Copyright: © 2022, Cornell University. All rights reserved.

PY - 2022

Y1 - 2022

N2 - Representing a sparse histogram, or more generally a sparse vector, is a fundamental task in differential privacy. An ideal solution would require space close to information-theoretical lower bounds, have an error distribution that depends optimally on the desired privacy level, and allow fast random access to entries in the vector. However, existing approaches have only achieved two of these three goals. In this paper we introduce the Approximate Laplace Projection (ALP) mechanism for approximating k-sparse vectors. This mechanism is shown to simultaneously have information-theoretically optimal space (up to constant factors), fast access to vector entries, and error of the same magnitude as the Laplace mechanism applied to dense vectors. A key new technique is a unary representation of small integers, which we show to be robust against “randomized response” noise. This representation is combined with hashing, in the spirit of Bloom filters, to obtain a space-efficient, differentially private representation. Our theoretical performance bounds are complemented by simulations showing that the constant factors on the main performance parameters are quite small and supporting practicality of the technique.

AB - Representing a sparse histogram, or more generally a sparse vector, is a fundamental task in differential privacy. An ideal solution would require space close to information-theoretical lower bounds, have an error distribution that depends optimally on the desired privacy level, and allow fast random access to entries in the vector. However, existing approaches have only achieved two of these three goals. In this paper we introduce the Approximate Laplace Projection (ALP) mechanism for approximating k-sparse vectors. This mechanism is shown to simultaneously have information-theoretically optimal space (up to constant factors), fast access to vector entries, and error of the same magnitude as the Laplace mechanism applied to dense vectors. A key new technique is a unary representation of small integers, which we show to be robust against “randomized response” noise. This representation is combined with hashing, in the spirit of Bloom filters, to obtain a space-efficient, differentially private representation. Our theoretical performance bounds are complemented by simulations showing that the constant factors on the main performance parameters are quite small and supporting practicality of the technique.

KW - Algorithms

KW - differential privacy

KW - histograms

KW - sparse vectors

UR - http://www.scopus.com/inward/record.url?scp=85140973339&partnerID=8YFLogxK

U2 - 10.29012/jpc.809

DO - 10.29012/jpc.809

M3 - Journal article

AN - SCOPUS:85140973339

VL - 12

JO - Journal of Privacy and Confidentiality

JF - Journal of Privacy and Confidentiality

SN - 2575-8527

IS - 2

ER -