# A Linear Reduction Method for Local Differential Privacy and Log-lift

This paper considers the problem of publishing data X while protecting correlated sensitive information S. We propose a linear method to generate the sanitized data Y with the same alphabet 𝒴 = 𝒳 that attains local differential privacy (LDP) and log-lift at the same time. It is revealed that both LDP and log-lift are inversely proportional to the statistical distance between conditional probability P_Y|S(x|s) and marginal probability P_Y(x): the closer the two probabilities are, the more private Y is. Specifying P_Y|S(x|s) that linearly reduces this distance |P_Y|S(x|s) - P_Y(x)| = (1-α)|P_X|S(x|s) - P_X(x)|,∀ s,x for some α∈ (0,1], we study the problem of how to generate Y from the original data S and X. The Markov randomization/sanitization scheme P_Y|X(x|x') = P_Y|S,X(x|s,x') is obtained by solving linear equations. The optimal non-Markov sanitization, the transition probability P_Y|S,X(x|s,x') that depends on S, can be determined by maximizing the data utility subject to linear equality constraints. We compute the solution for two linear utility function: the expected distance and total variance distance. It is shown that the non-Markov randomization significantly improves data utility and the marginal probability P_X(x) remains the same after the linear sanitization method: P_Y(x) = P_X(x), ∀ x ∈𝒳.

READ FULL TEXT