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2. (10 points) In class, we discussed how to represent XOR-like functions using quadratic features, since standard linear classifiers (such as perceptrons) are insufficient for this task. However, here we show that XOR-like functions can indeed be simulated using multi-layer networks of perceptrons. This example shows a glimpse of the expressive power of "deep neural networks": merely increasing the depth from 1 to 2 layers can help reproduce nonlinear decision boundaries. a. Consider a standard two-variable XOR function, where we have 2-dimensional inputs - 1 if x1 = x2 x1,22 = 1, and output y = x1 (XOR)x = 1 otherwise Geometrically argue why a single perceptron cannot be used to simulate the above function. b. Graphically depict, and write down the perceptron equation for, the optimal decision region for the following logical functions: (i) x1 (AND) (NOT (x2)) (ii) (NOT(x1) ) (AND)x2 (iii) x1 (OR)x2 Make note of the weights corresponding to the optimal decision boundary for each function. c. Using the above information, simulate a two-layer perceptron network for the XOR operation with the learned weights from Part (b). You can do this by taking the outputs of the logical functions presented in Part (b) and combining them carefully.

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