Optimal Boolean Locality-Sensitive Hashing.

For $0 leq beta u003c alpha u003c 1$ the distribution $mathcal{H}$ over Boolean functions $h colon {-1, 1}^d {-1, 1}$ that minimizes the expression begin{equation*} rho_{alpha, beta} = frac{log(1/Pr_{substack{h sim mathcal{H} (x, y) text{ $alpha$-corr.}}}[h(x) = h(y)])}{log(1/Pr_{substack{h sim mathcal{H} (x, y) text{ $beta$-corr.}}}[h(x) = h(y)])} end{equation*} assigns nonzero probability only to members of the set of dictator functions $h(x) = pm x_i$.