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Representing a sparse histogram, or more generally a sparse vector, is afundamental task in differential privacy. An ideal solution would use spaceclose to information-theoretical lower bounds, have an error distribution thatdepends optimally on the desired privacy level, and allow fast random access toentries in the vector. However, existing approaches have only achieved two ofthese three goals.  In this paper we introduce the Approximate Laplace Projection (ALP) mechanismfor approximating k-sparse vectors. This mechanism is shown to simultaneouslyhave information-theoretically optimal space (up to constant factors), fastaccess to vector entries, and error of the same magnitude as theLaplace-mechanism applied to dense vectors. A key new technique is a unaryrepresentation of small integers, which we show to be robust against``randomized response'' noise. This representation is combined with hashing, inthe spirit of Bloom filters, to obtain a space-efficient, differentiallyprivate representation.  Our theoretical performance bounds are complemented by simulations which showthat the constant factors on the main performance parameters are quite small,suggesting practicality of the technique.

Differentially Private Sparse Vectors with Low Error, Optimal Space, and Fast Access