2 ci- and (2-Regularization Consider sample points X, , ,, . . . , xn Rd and associated values yi,y2. . . . ,y" R, an n xd design matrix X -[Xi that the sample data has been centered and whitened so that each feature has mean 0 and variance 1 and the features are uncorrelated; ie., XX = n. For this question, we will not use a fictitious dimension nor a bias term; our linear regression function will output zero for x = 0. Consider linear least-squares regression with regularization in the C1-norm, also known as Lasso. The Lasso cost function is X, ] and an n-vector y yn]. For the sake of simplicity, assume . . . where w Rd and > 0 is the regularization parameter. Let w*-arg min weRd J(w) denote the weights that minimize the cost function. In the following steps, we will show that whitened training data decouples the features, so that w) is determined by the ih feature alone (i.e., column i of the design matrix X), regardless of the other features. This is true for both Lasso and ridge regression. 1. We use the notation X., to denote column i of the design matrix X, which represents the ith feature. Write J(w) in the following form for appropriate functions g and f 2. If w. > 0, what is the value of w? 3. If w 0 is the regularization parameter. Let w*-arg min weRd J(w) denote the weights that minimize the cost function. In the following steps, we will show that whitened training data decouples the features, so that w) is determined by the ih feature alone (i.e., column i of the design matrix X), regardless of the other features. This is true for both Lasso and ridge regression. 1. We use the notation X., to denote column i of the design matrix X, which represents the ith feature. Write J(w) in the following form for appropriate functions g and f 2. If w. > 0, what is the value of w? 3. If w