Supermodular Locality Sensitive Hashes

In this work, we show deep connections between Locality Sensitive Hashability and submodular analysis. We show that the LSHablility of the most commonly analyzed set similarities is in one-to-one correspondance with the supermodularity of these similarities when taken with respect to the symmetric difference of their arguments. We find that the supermodularity of equivalent LSHable similarities can be dependent on the set encoding. While monotonicity and supermodularity does not imply the metric condition necessary for supermodularity, this condition is guaranteed for the more restricted class of supermodular Hamming similarities that we introduce. We show moreover that LSH preserving transformations are also supermodular-preserving, yielding a way to generate families of similarities both LSHable and supermodular. Finally, we show that even the more restricted family of cardinality-based supermodular Hamming similarities presents promising aspects for the study of the link between LSHability and supermodularity. We hope that the several bridges that we introduce between LSHability and supermodularity paves the way to a better understanding both of supermodular analysis and LSHability, notably in the context of large-scale supermodular optimization.