Practice assignment
Question: 1
Statement
Assume that are principal components corresponding to nonzero
eigenvalues of the -dimensional centered data points .
Statement 1: each can be written as a linear combination of .
Statement 2: each can be written as a linear combination of .
Options
(a)
Statement 1 is correct but statement 2 is incorrect.
(b)
Statement 1 is incorrect but statement 2 is correct.
(c)
Both statements are correct.
(d)
Both statements are incorrect.
Answer:
(c)
Solution
In the first week, we have seen that residues after iterations become zero that is
it implies that each can be written as a linear combination of .
We know that the eigenvectors of the covariance matrix are the principal components of the
dataset and by the definition of eigenvectors, we have
, That is ach can be written as a linear combination of .
Question: 2
Statement
A transformation mapping is defined as
Which of the following options are the same as for two points ?
Hint: Rather than doing the calculation, try to figure out the appropriate kernel function.
(a)
(b)
(c)
(d)
Answer:
(a), (b), (d)
Solution
It is easy to verify that
It shows that the polynomial kernel of degree three refers to the given transformation . And
since the dot product is commutative, we can check that options (a), (b), and (d) refer to the same
expression.
Therefore the correct answers are options (a), (b), and (d).
Question: 3
Question: 1
Statement
Assume that are principal components corresponding to nonzero
eigenvalues of the -dimensional centered data points .
Statement 1: each can be written as a linear combination of .
Statement 2: each can be written as a linear combination of .
Options
(a)
Statement 1 is correct but statement 2 is incorrect.
(b)
Statement 1 is incorrect but statement 2 is correct.
(c)
Both statements are correct.
(d)
Both statements are incorrect.
Answer:
(c)
Solution
In the first week, we have seen that residues after iterations become zero that is
it implies that each can be written as a linear combination of .
We know that the eigenvectors of the covariance matrix are the principal components of the
dataset and by the definition of eigenvectors, we have
, That is ach can be written as a linear combination of .
Question: 2
Statement
A transformation mapping is defined as
Which of the following options are the same as for two points ?
Hint: Rather than doing the calculation, try to figure out the appropriate kernel function.
(a)
(b)
(c)
(d)
Answer:
(a), (b), (d)
Solution
It is easy to verify that
It shows that the polynomial kernel of degree three refers to the given transformation . And
since the dot product is commutative, we can check that options (a), (b), and (d) refer to the same
expression.
Therefore the correct answers are options (a), (b), and (d).
Question: 3
