▁ ▁Since ▁the ▁members ▁of ▁ ▁are ▁independently ▁chosen ▁for ▁any ▁, ▁ ▁is ▁a ▁- sens itive ▁family . ▁ ▁App lications ▁ ▁L SH ▁has ▁been ▁applied ▁to ▁several ▁problem ▁domains , ▁including : ▁ ▁Near - dup licate ▁detection ▁Hier arch ical ▁clust ering ▁Gen ome - wide ▁association ▁study ▁Image ▁similarity ▁identification ▁Visual R ank ▁Gene ▁expression ▁similarity ▁identification ▁Audio ▁similarity ▁identification ▁Near est ▁neighbor ▁search ▁Audio ▁finger print ▁Digital ▁video ▁finger print ing ▁Phys ical ▁data ▁organization ▁in ▁database ▁management ▁systems ▁Training ▁fully ▁connected ▁neural ▁networks ▁ ▁Method s ▁ ▁Bit ▁sampling ▁for ▁Ham ming ▁distance ▁ ▁One ▁of ▁the ▁easiest ▁ways ▁to ▁construct ▁an ▁L SH ▁family ▁is ▁by ▁bit ▁sampling . ▁This ▁approach ▁works ▁for ▁the ▁Ham ming ▁distance ▁over ▁d - dimensional ▁vectors ▁. ▁Here , ▁the ▁family ▁ ▁of ▁hash ▁functions ▁is ▁simply ▁the ▁family ▁of ▁all ▁the ▁pro jections ▁of ▁points ▁on ▁one ▁of ▁the ▁ ▁coordinates , ▁i . e ., ▁, ▁where ▁ ▁is ▁the ▁th ▁coordinate ▁of ▁. ▁A ▁random ▁function ▁ ▁from ▁ ▁simply ▁selects ▁a ▁random ▁bit ▁from ▁ ▁the ▁input ▁point . ▁This ▁family ▁has ▁the ▁following ▁parameters : ▁, ▁. ▁ ▁Min - wise ▁independent ▁permut ations ▁ ▁Suppose ▁▁ ▁is ▁composed ▁of ▁subsets ▁of ▁some ▁ground ▁set ▁of ▁enumer able ▁items ▁ ▁and ▁the ▁similarity ▁function ▁of ▁interest ▁is ▁the ▁J acc ard ▁index ▁. ▁If ▁ ▁is ▁a ▁permut ation ▁on ▁the ▁indices ▁of ▁, ▁for ▁ ▁let ▁. ▁Each ▁possible ▁choice ▁of ▁