The price of privacy and the limits of LP decoding

This work is at theintersection of two lines of research. One line, initiated by Dinurand Nissim, investigates the price, in accuracy, of protecting privacy in a statistical database. The second, growing from an extensive literature on compressed sensing (see in particular the work of Donoho and collaborators [4,7,13,11])and explicitly connected to error-correcting codes by Candès and Tao ([4]; see also [5,3]), is in the use of linearprogramming for error correction. Our principal result is the discovery of a sharp threshhold ρ*∠ 0.239, so that if ρ < ρ* and A is a random m x n encoding matrix of independently chosen standardGaussians, where m = O(n), then with overwhelming probability overchoice of A, for all x ∈ Rn, LP decoding corrects ⌊ ρ m⌋ arbitrary errors in the encoding Ax, while decoding can be made to fail if the error rate exceeds ρ*. Our boundresolves an open question of Candès, Rudelson, Tao, and Vershyin [3] and (oddly, but explicably) refutesempirical conclusions of Donoho [11] and Candès et al [3]. By scaling and rounding we can easilytransform these results to obtain polynomial-time decodable random linear codes with polynomial-sized alphabets tolerating any ρ < ρ* ∠ 0.239 fraction of arbitrary errors. In the context of privacy-preserving datamining our results say thatany privacy mechanism, interactive or non-interactive, providingreasonably accurate answers to a 0.761 fraction of randomly generated weighted subset sum queries, and arbitrary answers on the remaining 0.239 fraction, is blatantly non-private.