1. There are three discriminant functions, corresponding to three different perceptrons, f. (X2, X2) = x2 x2, 12(x1, x2) = X2 - 2 and 3(x1,x2) = -X2 +1. A feature vector x is assigned to class 1 iff > 0,12 0 and f; 0. a) For each discriminant function, draw the corresponding hyper-plane. A hyper-plane divides the feature space into two half-spaces. For each hyper-plane, determine the positive half-space and the negative half-space. b) Implement each discriminant function by drawing the corresponding perceptron. c) Classify each one of the following feature vectors: X2 = [10]" x2 = [1 1.5]" x3 = [3 2.5]" 2. Consider a dataset with four observations (X1, X2, X3, x4. Each observation is represented by a two-dimensional feature vector x = [X1 X2!". Each feature vector belongs to one of two classes, C or Cz. The two feature vectors that belong to C. are x = [2 3.5]" and X2 = [2 0.5]". The two feature vectors that belong to C2 are x3 = [2 21" and X4 = [2.5 21" a) Are the two classes linearly separable? is it possible to use a linear discriminant function f(x1, x2) = W_ xz + WX2 + b to completely and correctly separate both classes? b) Consider the use of non-linear features [X1 X2 XX2 * x3l" so that the classifier is characterized by a non-linear discriminant function f(x1,x2) = -4x2 - 4x2 + x + x3 + 7. Draw a diagram for the architecture of the non-linear perceptron c) For the case of the non-linear perceptron, described in the preceding paragraph, does the perceptron separates correctly both classes? Justify your answer. 3. Consider the multi-layer perceptron (MLP) which is shown in the following Figure. The activation function for each neuron is the logistic function (v) = The synaptic weight of neurons at the hidden layer are w} = 1 1 and w} = 6 --The vector of synaptic weights of the only neuron at the output layer is given by w} = [0, -1,1". If y 0.5 then x E C2. 1+e 1 wi wit X1 () a) A feature vector x = [02]" is applied to the MLP. Which are the values of the transformed features at the outputs of the hidden layer yi, y?? vi wii wo X2 b) To which class does x belong to? 1