▁similarity ▁dig est ▁schemes ▁such ▁as ▁T L SH , ▁S s deep ▁and ▁S d hash . ▁ ▁An ▁implementation ▁of ▁T L SH ▁is ▁available ▁as ▁open - source ▁software . ▁ ▁Random ▁projection ▁ ▁The ▁random ▁projection ▁method ▁of ▁L SH ▁due ▁to ▁Mos es ▁Char ik ar ▁called ▁Sim Hash ▁( also ▁sometimes ▁called ▁ar cc os ) ▁is ▁designed ▁to ▁approximate ▁the ▁cos ine ▁distance ▁between ▁vectors . ▁The ▁basic ▁idea ▁of ▁this ▁technique ▁is ▁to ▁choose ▁a ▁random ▁hyper plane ▁( defined ▁by ▁a ▁normal ▁unit ▁vector ▁) ▁at ▁the ▁out set ▁and ▁use ▁the ▁hyper plane ▁to ▁hash ▁input ▁vectors . ▁ ▁Given ▁an ▁input ▁vector ▁ ▁and ▁a ▁hyper plane ▁defined ▁by ▁, ▁we ▁let ▁. ▁That ▁is , ▁▁ ▁depending ▁on ▁which ▁side ▁of ▁the ▁hyper plane ▁ ▁lies . ▁ ▁Each ▁possible ▁choice ▁of ▁ ▁defines ▁a ▁single ▁function . ▁Let ▁ ▁be ▁the ▁set ▁of ▁all ▁such ▁functions ▁and ▁let ▁ ▁be ▁the ▁uniform ▁distribution ▁once ▁again . ▁It ▁is ▁not ▁difficult ▁to ▁prove ▁that , ▁for ▁two ▁vectors ▁, ▁, ▁where ▁ ▁is ▁the ▁angle ▁between ▁ ▁and ▁. ▁ ▁is ▁closely ▁related ▁to ▁. ▁ ▁In ▁this ▁instance ▁has hing ▁produces ▁only ▁a ▁single ▁bit . ▁Two ▁vectors ' ▁bits ▁match ▁with ▁probability ▁proportional ▁to ▁the ▁cos ine ▁of ▁the ▁angle ▁between ▁them . ▁ ▁St able ▁distributions ▁ ▁The ▁hash ▁function ▁▁ ▁maps ▁a ▁d ▁dimensional ▁vector ▁ ▁onto ▁the ▁set ▁of ▁integers . ▁Each ▁hash ▁function ▁in ▁the