06 07 2023 E T LAB

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Course: Numerical Analysis for Machine Learning Prof. E. Miglio - July 6th 2023 Duration of the exam: 2.5 hours. Exercise 1 Consider the picture of the Duomo di Milano given in file duomo.jpg. import matplotlib.pyplot as plt import numpy as np from matplotlib.image import imread image_path = 'duomo.jpg' img = imread(image_path) A = np.mean(np.array(img, dtype = np.float64)/255, axis = -1) plt.imshow(A, cmap = 'gray') plt.axis('off') A.shape 1. Compute the SVD associated with the picture, by using the standard algorithm, that gives the exact (up to round-off errors) decomposition. Then, plot the trend w.r.t. i of the singular values σi . Then, repeat the computation by considering the randomized SVD algorithm of rank k = 25, 50, 100, 200 and 400 . Plot the approximate singular values together with the exact singular values. Comment what you see. 2. We now want to compress the image. • Use the exact SVD to perform image compression for rank k = 25 , 50, 100, 200 and 400. Plot the compressed image and compute the reconstruction error as a function of k (use the matrix p-norm with p = 2). • Repeat the same exercise by using the randomized SVD algorithm. • Repeat the same exercise by using the randomized SVD algorithm with a +50% oversampling, that is by increasing by 50% the number of columns in the matrix random P (round the quantity 1.5k to the closest integer). • Finally, plot the trend of the reconstruction error of the dataset as a function of k in the three cases. Comment on the results. 3. Evaluate the time needed to compute the SVD with the three approaches (SVD, randomized SVD, ran- domized SVD with oversampling) in the case k = 200. Exercise 2 Consider the following function (where x = (x, y)) f(x) = 5x2 − 6xy + 5y2 + 4x + 4y (1) 1. Plot the function. 2. Compute…