In this optional assignment we will experiment with random hyperplanes
for classification. Your program will take a dataset as input and
produce new features following the procedure below. The input is in
the same format as for previous assignments.

Input data matrix X: n rows, m columns
Input training labels Y
Input value of k

For i = 0 to k do:
	a. Create random vector w where each wj is uniformly sampled between -1 and 1.
	
	b. Let xj be our training data points. Determine the largest and smallest wTxj
	across all xj. Select w0 randomly between [smallest wTxj, largest wTxj].

	c. Project training data X (each row is datapoint xj) onto w. 
	Let projection vector zi be Xw + w0 (here X has dimensions n by m and w is m by 1).
	Append (1+sign(zi))/2 as new column to the right end of Z. Remember that zi is
	a vector and so (1+sign(zi))/2 is 0 if the sign is -1 and 1 otherwise.
	
	d. Project test data X' (each row is datapoint xj) onto w. 
	Let projection vector z'i be X'w. Append z'i as new column to the right end 
	of Z'.
	
1. Run linear SVM on Z and predict on Z'
2. Do values of k=10, 100, 1000, and 10000.
3. How does the error compare to liblinear on original data X and X' for each k?

Submit a document containing the error of linear SVM (cross-validated C) on the 
first split of each of the six datasets on the course website. Do this on the original 
data representation and the new representation for all values of k.

Submit your program that creates features and run LinearSVC (in Python scikit)
on the new training data and predicts on the new test data. In LinearSVC set the
max_iter parameter to 10000 so that we do a deep search.

If you are working in a group then only one student needs to submit but both names
need to be on the submission document so that we know who the team is. The other student
in the team can leave their submission directory empty.

Directories: /afs/cad/courses/ccs/f18/cs/675/001/<ucid>.
For example if your ucid is abc12 then copy your programs into
/afs/cad/courses/ccs/f18/cs/675/001/abc12.

Your completed program is due before midnight Dec 8th.

Usman