1. (iv), (vi)

2. 

gy <- function() {

   library(lme4)
   data(InstEval)
   ie <- InstEval
   
   # easier if IDs are 1,2,3,..., and ratings are numeric
   factorToNumToFactor <- function(f) as.factor(as.numeric(f))
   ie$s <- factorToNumToFactor(ie$s)
   ie$d <- factorToNumToFactor(ie$d)
   ie$y <- as.numeric(ie$y)
   
   library(recosystem)
   r <- Reco()
   # training set will be all of ie, but need in recosystem form
   ### next line was a fill-in
   ie.trn <- data_memory(ie[,1],ie[,2],ie[,7],index1=TRUE)
   
   ### next line was a fill-in
   r$train(ie.trn,opts=list(dim=20,nmf=TRUE))
   result <- r$output(out_memory(),out_memory())
   ### next 3 lines were fill-ins
   w <- result$P
   h <- t(result$Q)
   aout <- w %*% h
  
   # for a given row in ie, find the predicted rating
   findPred <- function(row) {
      row <- as.numeric(row)
      userID <- row[1]
      instID <- row[2]
   ### next line was a fill-in
      round(aout[userID,instID])
   }

   # get vector of predicted ratings for ie
   preds <- apply(ie,1,findPred)

   ### next line was a fill-in
   mean(preds == ie$y)

}

print(gy())  # 0.2904619


3. 

nmf <- function(a,w0,h0,nIters) {
   w <- w0; h <- h0
   nch <- ncol(h)
   nrw <- nrow(w)
   for (iter in 1:nIters) {
      for (j in 1:nch) {
         h[,j] <- lm(a[,j] ~ w-1)$coef
      }
      for (i in 1:nrw) {
         w[i,] <- lm(a[i,] ~ t(h)-1)$coef
      }
   }
   list(w=w, h=h)
}

a <- rbind(2:5,c(4,3,5,1),c(2,2,1,4),1:4)
w0 <- matrix(1:8,ncol=2) 
h0 <- matrix(1:8,nrow=2) 
print(nmf(a,w0,h0,2))

# $w
#           [,1]      [,2]
# [1,]  5.077570 10.828505
# [2,] -5.426055 -4.330852
# [3,]  5.597083 10.142626
# [4,]  4.716699  9.309399
# 
# $h
#            [,1]       [,2]      [,3]       [,4]
# [1,] -1.3990331 -1.2300503 -1.958519 -0.8837465
# [2,]  0.8750231  0.8565012  1.260554  0.8783982