Question

Write the criterion function J MSE for a 2-class MSE classifier. Then add an L2 regularizer term (similar to the L2 regularizer term for Ridge Regression). Give any restrictions on b. (b) Solve for the MSE solution using a pseudoinverse technique.

Answer 1

The 2-class MSE classifier criterion function with an L2 regularizer is J_MSE = (1/N) * sum((y_i - f(x_i))^2) + lambda * norm(w)^2. To find the MSE solution using a pseudoinverse technique, solve the matrix equation w = inv(XtX + lambda * I) * XtY, ensuring the matrix is invertible or using the Moore-Penrose pseudoinverse as an alternative.

Step-by-step explanation:

The criterion function for a 2-class mean squared error (MSE) classifier is given by J_MSE = (1/N) * sum((y_i - f(x_i))^2), where N is the number of samples, y_i is the true label, and f(x_i) is the predicted label from the classifier. To add an L2 regularizer term, which is similar to the L2 regularizer term for Ridge Regression, the function becomes J_MSE = (1/N) * sum((y_i - f(x_i))^2) + lambda * norm(w)^2, where lambda is the regularization parameter and w represents the weight vector of the classifier. The regularized term helps to prevent overfitting by penalizing large weights.

The MSE solution can be solved using the pseudoinverse technique by setting the derivative of J_MSE with respect to w to zero and solving for w:

• Set up the matrix equation XtXw - XtY + lambda * I * w = 0, where X is the data matrix, Y is the label vector, I is the identity matrix, and lambda is the regularization parameter.
• Solve for w: w = inv(XtX + lambda * I) * XtY, where inv() denotes the matrix inverse and Xt is the transpose of X.

The pseudoinverse technique assumes that the matrix (XtX + lambda * I) is invertible, and if it's not, a computational technique such as Moore-Penrose pseudoinverse is used to find a solution.