06 09 2023 E T LAB

Pagina 1

Course: Numerical Analysis for Machine Learning Prof. E. Miglio - September 6th 2023 Duration of the exam: 2.5 hours. Exercise 1 Load the image of the Karman vortices with the following commands from matplotlib.image import imread import matplotlib.pyplot as plt import numpy as np import os plt.rcParams['figure.figsize'] = [16,8] A =imread(os.path.join('.','Karman_vortex.jpg')) X = np.mean(A,-1); # Convert RGB to grayscale img = plt.imshow(X) img.set.cmap('gray') plt.show() 1. Compute the economy SVD. 2. Let X be the matrix representing the true image and ˜X the approximation of rank r obtained using the SVD. Compute and plot the relative reconstruction error of the truncated SVD in the Frobenius norm as a function of the rank r. The expression of the relative reconstruction error is given by: ∥X − ˜X∥F ∥X∥F . 3. Square this error (and plot it) to compute the fraction of the missing variance as a function of r. 4. Find the rank ˜r for which the reconstruction captures 99% of the total variance. Exercise 2 Generate 100 artificial data points ( xi, yi) where xi is randomly generated in the interval [0 , 1] and yi = cos(4πxi) +ϵ; ϵ is a random noise in the interval [−0.2, 0.2]. Implement the SGD method to solve the regression problem for the data you have generated. Use an initial constant learning rate η = 0.001 an train a polynomial of the form hc = c0 + c1x + c2x2 + . . . + cnxn using your data ( n is the maximum degree of the polynomial). Assume that all the initial parameters ci are randomly generated in [ −0.4, 0.4]. Try different values of n. Try also different values for α to speed up the learning process. Plot the various results and comment them. Exercise 3 Consider a sigmoid neuron with 1D input x, weight w, bias b and output y = σ(wx + b). The target is the…