Question 1

Figure from the original question paper Two 5 x 5 binary images (digits 2 and 1, white = 1, black = 0) with text asking whether they are linearly separable for an MP neuron after flattening to 25 x 1 Consider the data points $\left(x_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, x_2 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}, x_3 = \begin{bmatrix} -1 \\ 1 \end{bmatrix}, x_4 = \begin{bmatrix} -1 \\ -1 \end{bmatrix}\right)$. Of these, the points ($x_1, x_2$ and $x_3$) belong to positive (1) class and the point $x_4$ belongs to negative (0) class. The perceptron uses the following decision rule $$\hat{y} = \begin{cases} 1, & \text{if } w^T x \geq 0 \\ 0, & w^T x < 0 \end{cases}$$ Based on the above data, answer the given subquestions. Suppose that the decision boundary passes through the origin and through the point $D = \begin{bmatrix} 0.5 \\ 1 \end{bmatrix}$. The weight vector is initialized to $w = \begin{bmatrix} w_0 \\ w_1 \end{bmatrix}$ with $w_0 = -1$. What is the value of $w_1$?