Reference no: EM133550368
Question 1. Let A be an m × d matrix, and let X = AAT . Assume that X has d distinct, non-zero eigenvalues. Assume that m » d. In order to find the eigendecomposition of X, we will need to find the eigendecomposition of an m × m matrix. Since m is much larger than d, this is slow. Give an algorithm for finding the eigenvectors and eigenvalues of X that only requires computing the eigendecomposition of a d × d matrix. You can use simple matrix operations and assume that you have an eigendecomposition "black box" subroutine, but avoid using the SVD as a black box.
Question 2. In this problem we explore some relationships between SVD, PCA and linear regression.
(a) True or false: linear regression is primarily a technique of supervised learning, i.e. where we are trying to fit a function to labeled data.
(b) True or false: PCA is primarily a technique of unsupervised learning, i.e. where we are trying to find structure in unlabeled data.
(c) True or false: SVD is primarily an operation on a dataset whereas PCA is primarily an operation on a matrix.
(d) A common problem in linear regression is multicollinearity, where the input variables are themselves linearly dependent. For example, imagine a healthcare data set where height is measured both in inches and centimetres. This is a problem because there may now be multiple w satisfying y = w · x. Explain how you could use a preprocessing step to solve this problem.
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