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1.[Analytical question] Consider two Normally distributed ra...

1.[Analytical question] Consider two Normally distributed random variables Y1and Y2 with expected values μ1 and μ2, variances σ12 and σ2, and correlation ρ. (a) State the joint probability distribution of these random variables. State it twice: once in a non-matrix and the se...

2. (10 points) In class, we discussed how to represent XOR-l...

2. (10 points) In class, we discussed how to represent XOR-like functions using quadratic features, since standard linear classifiers (such as perceptrons) are insufficient for this task. However, here we show that XOR-like functions can indeed be simulated using multi-layer networks of perceptro...

1. (10 points) Suppose we are given real-valued scalar data ...

1. (10 points) Suppose we are given real-valued scalar data (i.e., d = 1) belonging to one of two classes. We are given a set of three data samples with negative labels, X- = {0, 1, - -1}, , and a set of three data samples with positive labels, X+ = {-3,3,-2}. - Our goal is to build a classifier ...

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-...

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