## function soft-thresholding
soft_th <- function(X, lambda){
# assume X is a vector
result <- X
index1 <- which(X > lambda)
result[index1] <- X[index1] - lambda
index2 <- which(X < -lambda)
result[index2] <- X[index2] + lambda
index0 <- which(X <= lambda & X >= -lambda)
result[index0] <- 0
return(result)
}
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