tracefht_liu <-
function(A,k,trace.full=sum(A^2)){
m<-NROW(A)
mbig<-2^ceiling(log2(m))
R<-sample(c(-1,1),m,replace=TRUE)
Astar<-A*R
idx<-sample(mbig,k)
AOmega<-fht(Astar)[idx,]
AAOmega<-tcrossprod(AOmega,A)
A3Omega<-AAOmega%*%A
A4Omega<-tcrossprod(A3Omega,A)
tr<-sum(rowSums(AOmega*AOmega))/k
trAA2<-sum(rowSums(AAOmega*AAOmega))/k
trAA3<-sum(rowSums(A3Omega*A3Omega))/k
trAA4<-sum(rowSums(A4Omega*A4Omega))/k
c(trAA2*(trace.full/tr)^2, trAA3*(trace.full/tr)^3,trAA4*(trace.full/tr)^4)
}
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