1. The decision boundary in any binary classification proble...

1. The decision boundary in any binary classification problem can be intuitively viewed as a line/surface that partitions the test data points into two parts, one for each class. In this problem, we explore decision boundaries for k-nearest neighbors. a. Let us suppose that k = 1 and we are using...

1. (10 points) In class, we discussed the Lasso regularizati...

1. (10 points) In class, we discussed the Lasso regularization method, where the regularizer was chosen to be the l1-norm. Unfortunately, the optimal solution for this problem cannot be computed in closed form. However, we can derive an analytical closed form expression for the minimizer of a sli...

1. (10 points) Below, L(w) : Rd R are loss functions. For ea...

1. (10 points) Below, L(w) : Rd R are loss functions. For each L, (i) write down the gradient, and (ii) write down the closed form of the minimizer by setting the gradient to zero. a. = - b. L (w) = Il Aw - b11, where A is an orthonormal basis, i.e., its columns are unit norm and mutually perpen...

Instructions for the assignment In this assignment, we have...

Instructions for the assignment In this assignment, we have programming and writing questions. The Q4 is bonus for both undergraduate and graduate students. To switch between cell for code and for markdown, see the menu -> Cell -> Cell Type You could directly type the Latex equatio...

3. Implementing EM algorithm for MNIST dataset. Implement t...

3. Implementing EM algorithm for MNIST dataset. Implement the EM algorithm for fitting a Gaussian mixture model for the MNIST dataset. We reduce the dataset to be only two cases, of digits "2" and "6" only. Thus, you will fit GMM with C = 2. Use the data file d...

The resulting optimal pattern x from the equation argmin |...

The resulting optimal pattern x from the equation argmin | where train is the training dataset, and Xt the unseen/query pattern, is used for x € D, train which learning purpose? Used in multi-layer feed-forward networks to train the query pattern xt, and find its output x. Used in memory-...

Introduction to Machine Learning 1. Exercise 8.5 (e-Chap:8-...

Introduction to Machine Learning 1. Exercise 8.5 (e-Chap:8-11) in LFD. Show that the matrix Q described in the linear hard-margin SVM algorithm above is positive semi-definite (that is uT Qu ≥ 0 for any u). Note: The result means that the QP-problem is convex. Convexity is useful because this ...

1. (75 points) This question is related to Principle Compone...

1. (75 points) This question is related to Principle Component Analysis (PCA). You are to apply data pre-processing techniques to a collection of handwritten digit images t. You can load the whole dataset into Python using the function loadmat in Scipy.io. The matrix A contains all the images of si...

Introduction to Machine Learning 1. (20 points) Support Vec...

Introduction to Machine Learning 1. (20 points) Support Vector Machines. Given two data points x1 = (1; 0)T , y1 = 1, and x2 = (3; 0)T , y2 = 1. (a) Compute the optimal w and b in support vector machine by solving the primal formu- lation given as follows: min w;b 1 2wTw subject to yi(w...

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