CPS 290 : Algorithmic Foundations of Data Science February 3 , 2017 Lecture 6 : Dimensionality Reduction

which simply counts the number of coordinates where the points differ. Consider now the following simple hash family: H = {hk | hk(~x) = k bit of ~x} Such a hash function maps each item to one of two buckets. Let Zij denote the random variable that is d if items xi and xj map to different buckets and 0 otherwise, where the randomness is over the hash function chosen from H. Then it is easy to check that