Locality sensitive hash functions based on concomitant rank order statistics

Locality Sensitive Hash functions are invaluable tools for approximate near neighbor problems in high dimensional spaces. In this work, we are focused on LSH schemes where the similarity metric is the cosine measure. The contribution of this work is a new class of locality sensitive hash functions for the cosine similarity measure based on the theory of concomitants, which arises in order statistics. Consider <i>n</i> i.i.d sample pairs, {(<i>X</i>₁; <i>Y</i>₁); (<i>X</i>₂; <i>Y</i>₂); : : : ;(<i>X</i>_<i>n</i>; <i>Y</i>_<i>n</i>)} obtained from a bivariate distribution <i>f</i>(<i>X, Y</i>). Concomitant theory captures the relation between the order statistics of <i>X</i> and <i>Y</i> in the form of a rank distribution given by Prob(Rank(<i>Y</i>_i)=<i>j</i>-Rank(Xi)=<i>k</i>). We exploit properties of the rank distribution towards developing a locality sensitive hash family that has excellent collision rate properties for the cosine measure. The computational cost of the basic algorithm is high for high hash lengths. We introduce several approximations based on the properties of concomitant order statistics and discrete transforms that perform almost as well, with significantly reduced computational cost. We demonstrate the practical applicability of our algorithms by using it for finding similar images in an image repository.