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![Consider the two-dimensional XOR dataset D which contains the following 4 examples with binary
labels D ([0,0], −1), ([1, 1], −1), ([0, 1], +1), ([1,0], +1), i.e., the truth table of the logical XOR
function.
x1
22
y
0
0
-1
0
1
+1
1
0
+1
1
1
-1
⚫ Q7.1 Prove that D is not linearly separable. (Hint: Show that there cannot be a vector of
parameters w such that wTx > 0 for all examples x that are positive, and wTx < 0 for all
examples x that are negative. Do not forget to add the bias feature 20 = 1 to each example
x.)
•
Q7.2 Add a third feature to each example in the dataset that is the product of the two
original features, i.e. for each example x the new feature is computed as 23 = 21x2. Is the
new dataset linearly separable? Prove your answer.](https://content.bartleby.com/qna-images/question/6f56d031-b647-4796-9490-6bff54b90583/395b3776-775d-4a93-aa39-f94abc8e92af/opcu2ew_processed.png)
Transcribed Image Text:Consider the two-dimensional XOR dataset D which contains the following 4 examples with binary
labels D ([0,0], −1), ([1, 1], −1), ([0, 1], +1), ([1,0], +1), i.e., the truth table of the logical XOR
function.
x1
22
y
0
0
-1
0
1
+1
1
0
+1
1
1
-1
⚫ Q7.1 Prove that D is not linearly separable. (Hint: Show that there cannot be a vector of
parameters w such that wTx > 0 for all examples x that are positive, and wTx < 0 for all
examples x that are negative. Do not forget to add the bias feature 20 = 1 to each example
x.)
•
Q7.2 Add a third feature to each example in the dataset that is the product of the two
original features, i.e. for each example x the new feature is computed as 23 = 21x2. Is the
new dataset linearly separable? Prove your answer.
Transcribed Image Text:Let w be the current vector of parameters in the Perceptron algorithm, right before executing the
if statement on line 4 (the algorithm is on page 4 of Neural Networks slides), and t is the label
for Xn-
Q8.1 Prove that if the algorithm executes the if clause on line 5, i.e. the model made a
mistake on example xn, then it is true that twTx ≤0.
⚫ Q8.2 Let w be the new weight vector after executing the if clause on line 5, i.e. w = w+tnxn.
Prove that twxn≥tnwTxn-
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