CS331: Machine Learning Prof. Dr. Volker Roth [email protected] FS 2012 Melanie Rey [email protected] Department of Mathematics and Computer Science Bernoullistrasse 16 4056 Basel Exercise 2: Sample generation Date: Monday, March 12th 2012 In this exercise we implement some functions for generating artificial data to test our learning algorithms. Please upload you matlab code and the images you created with your code (exercise 2.3 below) to courses. 2.1: Generator Implement two different generators that generate n random vectors x ∈ Rd . • A generator gen uniform(n, d, a) that generates n uniformly distributed vectors in the hyper-cube [−a, a]d . The resulting distribution obtained from calling gen uniform(50, 2, 5) should look like the one depicted in Figure 1a • A generator gen gaussmix(n, d, µ1 , µ2 , σ12 , σ22 ), that corresponds to a mixture of two Gaussians G1 ∼ N (µ1 , σ12 Id ) and G2 ∼ N (µ2 , σ22 Id ). For each xi , sample with probability 0.5 from G1 and with probability 0.5 from G2 . The resulting distribution obtained from calling gen gaussmix(50, 2, (−5, −5)T , (5, 5)T , 1, 1) should look like the one depicted in Figure 1b. (a) Uniform (b) Mixture of Gaussians Figure 1: Examples of generated points 2.2: Supervisor Implement the following supervisor functions, that generates labels for the inputs x: 1 CS331: Machine Learning FS 2012 (a) 1D Regression (b) 2D Classification Figure 2: Visualization of a 1D regression example and a 2D classification example. • Linear supervisor: sup linear(x, w, σ 2 ) = hw, xi = wT x + where w is a parameter. • Sine supervisor: sup sin(x, w, σ 2 ) = sin(hw, xi) + where w is a parameter. • Ball supervisor: sup ball(x, r, σ 2 ) = kxk − r + , where r ∈ R is a parameter. Here, ∼ N (0, σ 2 ) is a noise term. To obtain binary labels for classification tasks, you can simple apply the sign function  −1 if y < 0 sgn(y) = 1 otherwise on the result of the supervisor. 2.3: Visualization Implement two functions that let you visualize the results for 1D and 2D case. The first function should let you plot a 1D sample {(x1 , y1 ), . . . , (xn , yn )} with xi , yi ∈ R together with a function f (e.g. the target function fρ or the learning machine’s function fS ). An example is given in Figure 2a. The second function is used to visualize the results from a classification task in 2D. It should let you plot the 2D sample {(x1 , y1 ), . . . , (xn , yn )} with xi ∈ R2 , yi ∈ R, together with a binary classification function θ : X → {−1, 1}. The easiest way to visualize the classification result is to evaluate θ on a grid. An example is given in Figure 2b. Use this visualization to illustrate the effect of the different generators and supervisors. 2