▁is ▁a ▁" learning ▁parameter ." ▁ ▁For ▁the ▁case ▁of ▁training ▁the ▁linear ▁weights , ▁, ▁the ▁algorithm ▁becomes ▁ ▁in ▁the ▁un normal ized ▁case ▁and ▁ ▁in ▁the ▁normal ized ▁case . ▁ ▁For ▁local - linear - arch itect ures ▁gradient - des cent ▁training ▁is ▁ ▁Pro jection ▁operator ▁training ▁of ▁the ▁linear ▁weights ▁ ▁For ▁the ▁case ▁of ▁training ▁the ▁linear ▁weights , ▁ ▁and ▁, ▁the ▁algorithm ▁becomes ▁ ▁in ▁the ▁un normal ized ▁case ▁and ▁ ▁in ▁the ▁normal ized ▁case ▁and ▁ ▁in ▁the ▁local - linear ▁case . ▁ ▁For ▁one ▁basis ▁function , ▁projection ▁operator ▁training ▁reduces ▁to ▁Newton ' s ▁method . ▁ ▁Ex amples ▁ ▁Log istic ▁map ▁ ▁The ▁basic ▁properties ▁of ▁radial ▁basis ▁functions ▁can ▁be ▁illustrated ▁with ▁a ▁simple ▁mathematical ▁map , ▁the ▁log istic ▁map , ▁which ▁maps ▁the ▁unit ▁interval ▁onto ▁itself . ▁It ▁can ▁be ▁used ▁to ▁generate ▁a ▁convenient ▁prototype ▁data ▁stream . ▁The ▁log istic ▁map ▁can ▁be ▁used ▁to ▁explore ▁function ▁approximation , ▁time ▁series ▁prediction , ▁and ▁control ▁theory . ▁The ▁map ▁origin ated ▁from ▁the ▁field ▁of ▁population ▁dynamics ▁and ▁became ▁the ▁prototype ▁for ▁cha otic ▁time ▁series . ▁The ▁map , ▁in ▁the ▁fully ▁cha otic ▁regime , ▁is ▁given ▁by ▁ ▁where ▁t ▁is ▁a ▁time ▁index . ▁The ▁value ▁of ▁x ▁at ▁time ▁t + 1 ▁is ▁a ▁par abol ic ▁function ▁of ▁x ▁at ▁time ▁t . ▁This ▁equation ▁represents ▁the ▁underlying ▁geometry ▁of ▁the