Description
Overview
In this assignment you will explore regression techniques on high-dimensional data. You will use a few data sets for this assignment:
• http://www.cs.utah.edu/˜jeffp/teaching/cs5140/A6/X.csv
• http://www.cs.utah.edu/˜jeffp/teaching/cs5140/A6/y.csv
• http://www.cs.utah.edu/˜jeffp/teaching/cs5140/A6/M.csv
• http://www.cs.utah.edu/˜jeffp/teaching/cs5140/A6/W.csv
For python, you can use the following approach to load the data:
X = np.loadtxt(’X.csv’, delimiter=’,’) y = np.loadtxt(’y.csv’, delimiter=’,’)
1 Linear Regression & Cross-Validation (100 points)
We will find coefficients alpha to estimate X*alpha ≈ y, using the provided datasets X and y. We will compare two approaches least squares and ridge regression. (e.g., in python as)
Least Squares: Set alpha = LA.inv(X.T @ X) @ X.T @ y.T
Ridge Regression: Set alphas = LA.inv(X.T @ X + s*np.identity(50)) @ X.T @ y.T
A (30 points): Solve for the coefficients alpha (or alphas) using Least Squares and Ridge Regression with s ∈ {0.2,0.4,0.8,1.0,1.2,1.4,1.6} (i.e. s will take on one of those 7 values each time you try, say obtaining alpha04 for s = 0.4). For each set of coefficients, report the error in the estimate yˆ of y as norm(y – X*alpha,2).
B (30 points): Create three row-subsets of X and Y
• X1 = X[:66,:] and Y1 = Y[:66]
• X2 = X[33:,:] and Y2 = Y[33:]
• X3 = np.vstack((X[:33,:], X[66:,:])) and Y3 = np.vstack((Y[:33], Y[66:]))]
Repeat the above procedure on these subsets and cross-validate the solution on the remainder of X and Y. Specifically, learn the coefficients alpha using, say, X1 and Y1 and then measure np.norm(Y[66:]
– X[66:,:] @ alpha,2).
C (15 points): Which approach works best (averaging the results from the three subsets): Least Squares, or for which value of s using Ridge Regression?
D (15 points): Use the same 3 test / train splits, taking their average errors, to estimate the average squared error on each predicted data point.
What is problematic about the above estimate, especially for the best performing parameter value s?
E (10 points): Even circumventing the issue raised in part D, what assumptions about how the data set (X,y) is generated are needed in an assessment based on cross-validation?
2 Bonus: Matching Pursuit (5 points)
Consider a linear equation W = M*S where M is a measurement matrix filled with random values {−1,0,+1} (although now that they are there, they are no longer random), and W is the output of the sparse signal S when measured by M.
Use Matching Pursuit (as described in the book as Algorithm 5.5.1) to recover the non-zero entries from S. Record the order in which you find each entry and the residual vector after each step.
