DESCRIPTION

Students:  Please keep in mind the OMSI rules.  Save your files often,
make sure OMSI fills your entire screen at all times, etc.  Remember
that clicking CopyQtoA will copy the entire question box to the answer
box.

In questions involving code which will PARTIALLY be given to you in the
question specs, you may need add new lines.  There may not be
information given as to where the lines should be inserted.

MAKE SURE TO RUN THE CODE IN PROBLEMS THAT INVOLVE CODE!

QUESTION -ext .R -run 'Rscript ./omsi_answer1.R'

(R code answer, 20 points.) One of the first topics usually covered
in a linear algebra course is formulating a system of linear equations
in matrix form.  For example,

3u + 4v = 18
u - v = -1

is expressed as Ax = y, where A = rbind(c(3,4),c(1,-1)), x is the column
vector c(u,v) and y is the column vector c(18,-1).  We find the x vector
via solve(A) %*% y.

But the system may have redundant equations, say

3u + 4v = 18
u - v = -1
4u + 3v = 17

Working by hand, we would simply delete one equation.  But
programmatically, we can use a generalized inverse to solve this or any
other redundant system.  Do so for the above example.

A <- rbind(                         )
y <- c(               )
print('the solution is:')

QUESTION -ext .R -run 'Rscript ./omsi_answer2.R'

(R code answer, 40 points)  Use softImpute() on the House Voting
data, omitting the party column.  Use rank 5, and (very important)
convert the data to 'Incomplete' class first.  Congressperson 3 did not
vote on Bill 1.  Find the predicted vote (raw estimated value, not
rounded to 0 or 1).

set.seed(1)
library(rectools)
library(softImpute)

QUESTION -ext .R -run 'Rscript ./omsi_answer3.R'

(R code answer, 40 points)  Implement the ALS code following
(6.46), with the call form

   als(m,r,nmf=FALSE,
      w0=matrix(runif(nrow(m)*r),nrow=nrow(m)),
      eps=0.01*max(abs(m)),maxIters=100) 

Here ALS will be applied to a matrix m, producing rank-r factors.  The
matrix m is assumed completely intact, no NAs.  If nmf is TRUE, use
Nonnegative Matrix Factorization, by replacing any negative values in
the factors at each iteration. w0 is the initial guess for the matrix w. 
Iteration will proceed until the change in Frobenius norm between 
consecutive w %*% h values is less than eps, but will stop if the 
number of iterations reaches maxIters.


library(MASS)

als <- function(m,r,nmf=FALSE,
   w0=matrix(runif(nrow(m)*r),nrow=nrow(m)),
   eps=0.01*max(abs(m)),maxIters=100) 
{
   w <- w0
   h <- matrix(rep(0,r*(ncol(m))),nrow=r)

   for (iter in 1:maxIters) {

      for (j in 1:ncol(m)) {
         h[,j] <- 
         if (nmf) 
      }

      for (i in 1:nrow(m)) {
         w[i,] <- 
         if (nmf)
      }

      if (                     ) break
      if (iter == maxIters) warning(            )
   }
   list(w=w,h=h,prd=w %*% h)
}

set.seed(999)
m <- matrix(sample(1:100,48),nrow=6)
res <- als(m,3)