Deep Learning - Week 1
Common data for questions 1,2 and 3
In the figure shown below, the blue points belong to class 1 (positive class) and the
red points belong to class 0 (negative class). Suppose that we use a perceptron model,
with the weight vector w as shown in the figure, to separate these data points. We
define the point belongs to class 1 if wT x ≥ 0 else it belongs to class 0.
y
C
−1 1
G
−1
1. The points G and C will be classified as?
Note: the notation (G, 0) denotes the point G will be classified as class-0 and (C, 1)
denotes the point C will be classified as class-1
(a) (C, 0), (G, 0)
(b) (C, 0), (G, 1)
(c) (C, 1), (G, 1)
(d) (C, 1), (G, 0)
Correct Answer: (d)
Solution:
0
w= ,
1.5
(
1 if wT x > 0
x∈
0 if wT x ≤ 0
For C(−0.6, 0.2):
T
−0.6
w x = 0 1.25 = (0)(−0.6) + (1.25)(0.2) = 0.25
0.25
∴ (C, 1)
For G(0.5, −0.5):
T
0.5
w x = 0 1.25 = (0)(0.5) + (1.25)(−0.5) = −0.625
−0.5
∴ (G, 0)
2. The statement that “there exists more than one decision lines that could separate
these data points with zero error” is,
(a) True
(b) False
Correct Answer: (a)
Solution: The given statement is True.
In the perceptron algorithm, when the data points are linearly separable, there can
exist multiple hyperplanes (decision lines) that perfectly classify the data points with
zero error. This is because a decision boundary depends on the orientation of the
separating hyperplane and the margin around it, which can vary as long as it satisfies
the linear separability condition.
For example, in the graph provided, multiple lines can separate the red and blue data
points such that all points are correctly classified. These decision boundaries can
differ in slope and position while still achieving zero classification error. Hence, the
solution is True.
3. Suppose that we multiply the weight vector w by −1. Then the same points G and
C will be classified as?
(a) (C, 0), (G, 0)
(b) (C, 0), (G, 1)
(c) (C, 1), (G, 1)
(d) (C, 1), (G, 0)
Correct Answer: (b)
Solution: Simply multiply w by −1 and repeat the calculations from question 1.
4. Which of the following can be achieved using the perceptron algorithm in machine
learning?
(a) Grouping similar data points into clusters, such as organizing customers based
on purchasing behavior.
(b) Solving optimization problems, such as finding the maximum profit in a business
scenario.
(c) Classifying data, such as determining whether an email is spam or not.
(d) Finding the shortest path in a graph, such as determining the quickest route
between two cities.
Correct Answer: (c)
Solution: Perceptron can only classify, linearly separable data.
5. Consider the following table, where x1 and x2 are features and y is a label
x1 x2 y
0 0 1
0 1 1
1 0 1
1 1 0
Assume that the elements in w are initialized to zero and the perception learning
algorithm is used to update the weights w. If the learning algorithm runs for long
enough iterations, then
(a) The algorithm never converges
(b) The algorithm converges (i.e., no further weight updates) after some iterations
(c) The classification error remains greater than zero
(d) The classification error becomes zero eventually
Correct Answer: (b),(d)
Solution: Since the data points are linearly separable, the algorithm converges, visu-
alize it using a graphing tool.
6. We know from the lecture that the decision boundary learned by the perceptron is a
line in R2 . We also observed that it divides the entire space of R2 into two regions,
suppose that the input vector x ∈ R4 , then the perceptron decision boundary will
divide the whole R4 space into how many regions?
(a) It depends on whether the data points are linearly separable or not.
(b) 3
(c) 4
(d) 2
(e) 5
Correct Answer: (d)
Solution: A line will become a hyperplane in R4 but still it will divide the region in
2 halves.
7. Choose the correct input-output pair for the given MP Neuron.
(
1, if x1 + x2 + x3 < 2
f (x) =
0, otherwise
(a) y = 1 for (x1 , x2 , x3 ) = (0, 0, 0)
(b) y = 0 for (x1 , x2 , x3 ) = (0, 0, 1)
(c) y = 1 for (x1 , x2 , x3 ) = (1, 0, 0)
(d) y = 1 for (x1 , x2 , x3 ) = (1, 1, 1)
(e) y = 0 for (x1 , x2 , x3 ) = (1, 0, 1)
Correct Answer: (a),(c),(e)
Solution: Substituting values into the above expression and evaluating them yields
the result.
8. Consider
the following table, where x1 and x2 are features (packed into a single vector
x
x = 1 ) and y is a label:
x2
x1 x2 y
0 0 0
0 1 1
1 0 1
1 1 1
Suppose that the perceptron model is used to classify
the data points. Suppose
1
further that the weights w are initialized to w = . The following rule is used for
1
classification,
(
1 if wT x > 0
y=
0 if wT x ≤ 0
The perceptron learning algorithm is used to update the weight vector w. Then, how
many times the weight vector w will get updated during the entire training process?
(a) 0
(b) 1
(c) 2
(d) Not possible to determine
Correct Answer: (a)
Solution: Upon computing wT x for all data points with the initial weight values, all
the points are correctly classified. Hence, update is not required.
9. Which of the following threshold values of MP neuron implements AND Boolean
function? Assume that the number of inputs to the neuron is 3 and the neuron does
not have any inhibitory inputs.
(a) 1
(b) 2
(c) 3
(d) 4
(e) 5
Correct Answer: (c)
Solution: suppose, we set θ = 4, then summing all the input never exceeds 3, therefore,
the neuron won’t fire, And suppose we set θ < 3 then it won’t satisfy the AND
operator.
−1
10. Consider points shown in the picture. The vector w = . As per this weight
1
vector, the Perceptron algorithm will predict which classes for the data points x1 and
x2 .
NOTE: (
1 if wT x > 0
y=
−1 if wT x ≤ 0
x1 (1.5, 2)
w
x
x2 (−2.5, −2)
(a) x1 = −1
(b) x1 = 1
(c) x2 = −1
(d) x2 = 1
Correct Answer: (b),(d)
−1
Solution: The decision boundary is wT x= 0. Hence for w = , anything in the
1
direction of w will have wT x > 0 and will get labelled 1.
