2 - We Are All Just Numbers Here...

Question 1

Who was William Rowan Hamilton?

Answer 1

An Irish mathematician known for his work on quaternions.

Question 2

What significant event happened on October 16, 1843?

Answer 2

Hamilton had a flash of inspiration for the quaternion formula while walking along the Royal Canal.

Question 3

What is the fundamental formula for quaternion multiplication?

Answer 3

i² = j² = k² = ijk = -1.

Question 4

What did Hamilton etch on the stone of Brougham Bridge?

Answer 4

The fundamental formula for quaternion multiplication.

Question 5

Define a scalar quantity.

Answer 5

A stand-alone number that represents magnitude only.

Question 6

Define a vector.

Answer 6

A quantity that has both magnitude and direction.

Question 7

What are the components of a vector?

Answer 7

The x-component and y-component.

Question 8

How can the magnitude of a vector be calculated?

Answer 8

Using the Pythagorean theorem: √(x² + y²).

Question 9

What does Newton’s Second Law of Motion state?

Answer 9

Acceleration is proportional to the force acting on an object and they have the same direction.

Question 10

What geometrical shape is used to represent vector addition?

Answer 10

A parallelogram.

Question 11

What is the resultant vector in the example of a man walking from (0,0) to (6,9)?

Answer 11

The vector from (0,0) to (6,9).

Question 12

What is the net distance in the xy coordinate space from the origin to (6,9)?

Answer 12

10.82 miles.

Question 13

What happens when you subtract vectors?

Answer 13

It indicates if one force is acting against another.

Question 14

What is the effect of multiplying a vector by a scalar?

Answer 14

It scales the vector’s magnitude.

Question 15

Define a unit vector.

Answer 15

A vector with a magnitude of 1.

Question 16

What is the dot product of two vectors?

Answer 16

The magnitude of one vector multiplied by the projection of another onto it.

Question 17

What does a dot product of zero indicate?

Answer 17

The two vectors are orthogonal (at right angles).

Question 18

How is the dot product calculated using vector components?

Answer 18

a.b = a1b1 + a2b2.

Question 19

What is the significance of Hamilton’s work on quaternions for machine learning?

Answer 19

It laid foundational mathematical concepts important for vector analysis.

Question 20

Fill in the blank: A ______ is a mathematical entity composed of four elements.

Answer 20

quaternion.

Question 21

True or False: The magnitude of a vector can be negative.

Answer 21

False.

Question 22

What does the projection of one vector onto another represent?

Answer 22

The ‘shadow cast’ by one vector onto another.

Question 23

What is the equation for the scalar quantity when dealing with vectors a and b?

Answer 23

a.b = a 1 b 1 + a 2 b 2

Question 24

What do the vectors i and j represent in the context of dot products?

Answer 24

Orthogonal vectors, where i.j and j.i are zero, and both i.i and j.j equal 1

Question 25

What does a perceptron output if the weighted sum of its inputs plus the bias term is greater than 0?

Answer 25

1

Question 26

What is the output of a perceptron if the weighted sum is less than or equal to 0?

Answer 26

-1

Question 27

In the perceptron model, how can the weights be represented?

Answer 27

As a vector w = (w1, w2)

Question 28

What geometrical concept does the perceptron use to separate data points into clusters?

Answer 28

A linearly separating hyperplane

Question 29

What is the relationship between the weight vector w and the separating hyperplane?

Answer 29

The vector w is orthogonal to the hyperplane

Question 30

What does the dot product of a data point vector and the weight vector indicate?

Answer 30

The distance of the data point from the hyperplane

Question 31

What happens when a data point lies on the hyperplane?

Answer 31

The dot product with the weight vector equals zero

Question 32

What is the significance of the bias term in a perceptron?

Answer 32

It moves the hyperplane away from the origin without changing its orientation

Question 33

Fill in the blank: The perceptron learning algorithm guarantees to find one separating hyperplane, but not necessarily the _____ one.

Answer 33

best

Question 34

What is the mathematical representation of a one-column matrix with two elements?

Answer 34

A column matrix indexed by numbers 1 and 2

Question 35

What is the process of flipping a column matrix on its side called?

Answer 35

Taking the transpose of a matrix

Question 36

What is the notation for the transpose of matrix A?

Answer 36

A T

Question 37

In the context of matrices, what is a vector?

Answer 37

A particular form of matrix with either one row or one column

Question 38

What is the relationship between the number of columns in the first matrix and the number of rows in the second for taking a dot product?

Answer 38

They must be equal

Question 39

How can the weighted sum of inputs in a perceptron be concisely written?

Answer 39

As the dot product w T x

Question 40

What does the perceptron learn from a set of input data vectors?

Answer 40

The weight vector that represents a hyperplane separating the data into two clusters

Question 41

What is the significance of the hyperplane in the context of classification?

Answer 41

It determines the classification of new data points based on their position relative to it

Question 42

True or False: The perceptron can classify data points as 'obese' or 'not-obese' based on their position relative to the hyperplane.

Answer 42

True

Question 43

What is the role of modern deep neural networks in relation to the perceptron?

Answer 43

They build upon the foundational concepts established by the perceptron

Question 44

What is a perceptron learning algorithm?

Answer 44

A computationally viable algorithm for binary classification that involves finding a hyperplane to separate data into two groups.

Question 45

What defines a 'solution' in the context of perceptrons?

Answer 45

A hyperplane that linearly separates the data into two groups.

Question 46

Who developed a significant proof regarding the perceptron learning algorithm in 1962?

Answer 46

Henry David Block.

Question 47

What did Block's proof establish?

Answer 47

Upper bounds for the number of mistakes made by the perceptron learning algorithm.

Question 48

What is the focus of Minsky and Papert's book 'Perceptrons'?

Answer 48

A class of computations that make decisions by weighing evidence.

Question 49

What was a notable criticism made by Block in his review of 'Perceptrons'?

Answer 49

He objected to Minsky and Papert's implication that cyberneticists should have known about earlier convergence proofs.

Question 50

What is the significance of the term 'cybernetics'?

Answer 50

The study of control and communication in the animal and the machine.

Question 51

What are the six variables used to categorize patients in the discussed pandemic scenario?

Answer 51

* x1 = age * x2 = body mass index * x3 = has difficulty breathing (yes = 1/no = 0) * x4 = has fever (yes/no) * x5 = has diabetes (yes/no) * x6 = chest CT scan (0 = clear, 1 = mild infection, 2 = severe infection)

Question 52

What does the outcome 'y' represent for each patient?

Answer 52

y = -1 (did not need ventilator support) or y = 1 (needed ventilator support).

Question 53

What is the goal of training a perceptron in this context?

Answer 53

To find a separating hyperplane for the data points.

Question 54

What is the first step in the perceptron training algorithm?

Answer 54

Initialize the weight vector to zero: set w = 0.

Question 55

What condition necessitates updating the weight vector in the perceptron algorithm?

Answer 55

If y w^T x ≤ 0.

Question 56

How does the perceptron determine if the weights are correct?

Answer 56

If the expression y w^T x is positive.

Question 57

What does the convergence proof by Minsky and Papert establish?

Answer 57

The perceptron will converge to a solution in a finite number of steps if one exists.

Question 58

What is the significance of the dot product of weight vectors during training?

Answer 58

It indicates how closely the weight vector aligns with the desired weight vector.

Question 59

What does the term 'XOR problem' refer to in perceptrons?

Answer 59

A problem that cannot be solved by a single layer of perceptrons, as it cannot be linearly separated.

Question 60

What is the relationship between lower and upper bounds in computational complexity?

Answer 60

Lower bounds indicate what is impossible, while upper bounds measure resource limits for solutions.

Question 61

What does the weight vector w represent in the perceptron model?

Answer 61

The parameters that define the hyperplane separating the data.

Question 62

Fill in the blank: The perceptron learning algorithm updates the weight vector by adding _______.

Answer 62

y x.

Question 63

True or False: The perceptron algorithm guarantees a solution for all types of data.

Answer 63

False.

Question 64

What major assumption is made about the data in the context of perceptrons?

Answer 64

The data are linearly separable.

Question 65

What is the bias term in the perceptron model denoted as?

Answer 65

w0.

Question 66

What is a perceptron?

Answer 66

A simple type of artificial neuron used in machine learning

Question 67

What problem did Minsky and Papert prove that a single layer of perceptrons could not solve?

Answer 67

The XOR problem

Question 68

What are the four data points involved in the XOR problem?

Answer 68

* (0, 0) * (1, 0) * (1, 1) * (0, 1)

Question 69

What must a perceptron output for the points (0, 0) and (1, 1)?

Answer 69

y = 1

Question 70

What must a perceptron output for the points (1, 0) and (0, 1)?

Answer 70

y = -1

Question 71

What does the term 'multi-layer perceptrons' refer to?

Answer 71

Perceptrons stacked such that the output of one feeds into the input of another

Question 72

What algorithm was published by Rumelhart, Hinton, and Williams in 1986?

Answer 72

Backpropagation

Question 73

What does the backpropagation algorithm rely on?

Answer 73

Calculus and optimization theory

Question 74

What is the significance of the year 1982 in neural network research?

Answer 74

A physicist's unique solution to a biological problem re-energized the field

Question 75

What is the goal of the perceptron algorithm?

Answer 75

To find a linearly separating hyperplane

Question 76

What is the initial step in the perceptron algorithm?

Answer 76

Initialize the weight vector to zero: set w = 0

Question 77

When does the weight vector get updated in the perceptron algorithm?

Answer 77

If y w^T x ≤ 0

Question 78

What is the equation for updating the weight vector in the perceptron?

Answer 78

w_new = w_old + y x

Question 79

What does γ (gamma) represent in the perceptron algorithm?

Answer 79

The distance between the linear separating hyperplane and the closest data point

Question 80

What is the dot product of a vector with itself always greater than or equal to?

Answer 80

0

Question 81

What happens to the dot product w^T w* after each update?

Answer 81

It grows by at least γ

Question 82

What happens to the dot product w^T w after each update?

Answer 82

It grows by at most 1

Question 83

How can the number of updates M required for the perceptron to converge be described?

Answer 83

M is always a finite quantity

Question 84

What is the maximum number of updates required for convergence in the perceptron algorithm?

Answer 84

1 over γ²

Question 85

True or False: The perceptron algorithm guarantees convergence in a finite number of steps.

Answer 85

True

Question 86

In the context of the perceptron, what does the term 'linearly separating hyperplane' refer to?

Answer 86

A hyperplane that separates different classes of data points

Question 87

What is a key limitation of a single layer perceptron?

Answer 87

It cannot solve problems like XOR that are not linearly separable

Question 88

What is the significance of normalization in the perceptron algorithm?

Answer 88

It ensures all input data points have magnitudes less than or equal to 1

Question 89

Fill in the blank: The perceptron will converge without fail in a finite number of _______.

Answer 89

steps