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MLF week 6 - real exam questions
28 questions pulled from past papers whose wording matches this week's topics, newest first, duplicates removed. Pick your answers, then press Check answers. Figures and answer keys are the originals.
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Comprehension passage for this question
Based on the above data, answer the given subquestions.
Let 
and 
denote the singular values of the matrix 
. Find the value of 
.
and denote the singular values of the matrix . Find the value of .Comprehension passage for this question
Consider the following dataset consisting of two features 
and 
:
We want to reduce the dimensionality of the dataset from 
to 
using the following encoder- decoder pairs.
The reconstruction loss is defined as the mean squared distance between the reconstructed input and the original input. Based on the above information, answer the given sub-questions.Compute the reconstruction loss for Pair 1. NOTE: Your answer should be an integer
Comprehension passage for this question
Consider the following dataset consisting of two features and :
We want to reduce the dimensionality of the dataset from to using the following encoder- decoder pairs.
The reconstruction loss is defined as the mean squared distance between the reconstructed input and the original input. Based on the above information, answer the given sub-questions.Compute the reconstruction loss for Pair 2. NOTE: Your answer should be an integer
Let 
be a real matrix whose largest singular value is 
. Let 
. Which of the following is true 
be a real matrix whose largest singular value is . Let . Which of the following is true - The largest singular value of
is 
. - The smallest singular value of is .
- The largest singular value of is
. - We don't have enough information to find the largest singular value of .
Inscript 
is a real matrix of shape 
with the following singular values: 
. Find the rank of .
is a real matrix of shape with the following singular values: . Find the rank of .Consider the matrix
. Find the sum of all the singular values of .
. Find the sum of all the singular values of .Comprehension passage for this question
Consider the following dataset in 
Run PCA on this dataset. Based on the above data, answer the given subquestions.If
is the covariance matrix of the dataset, find 
. Enter the nearest integer as your answer.
is the covariance matrix of the dataset, find . Enter the nearest integer as your answer.Comprehension passage for this question
Consider the following dataset in Run PCA on this dataset. Based on the above data, answer the given subquestions.Let 
be the unit-norm eigenvector corresponding to the smallest eigenvalue of the covariance matrix. Find the variance of the dataset along the direction . Enter the nearest integer as your answer.
be the unit-norm eigenvector corresponding to the smallest eigenvalue of the covariance matrix. Find the variance of the dataset along the direction . Enter the nearest integer as your answer.Comprehension passage for this question
If 
and 
, where
Based on the above data, answer the given subquestions.Find the covariance matrix of 
.
.
Consider two datasets in 
, 
and 
, each having three points as shown below. The figure is drawn to scale. PCA is run on both these datasets separately.
and 
are the covariance matrices of the two datasets respectively. If 
is the eigenvector corresponding to the largest eigenvalue of and 
is the eigenvector corresponding to the smallest eigenvalue of , find the absolute value of 
. Both eigenvectors have unit norm. Your answer should be an integer.
, and , each having three points as shown below. The figure is drawn to scale. PCA is run on both these datasets separately.
and are the covariance matrices of the two datasets respectively. If is the eigenvector corresponding to the largest eigenvalue of and is the eigenvector corresponding to the smallest eigenvalue of , find the absolute value of . Both eigenvectors have unit norm. Your answer should be an integer.Comprehension passage for this question
If and , where
Based on the above data, answer the given subquestions.Find the covariance matrix of .
.
Comprehension passage for this question
Based on the above data, answer the given subquestions.
The sample covariance matrix C for the given dataset is
Comprehension passage for this question
Based on the above data, answer the given subquestions.
The eigenvalues of the covariance matrix are
Comprehension passage for this question
Based on the above data, answer the given subquestions.
The first principal component is
Which of the following is true about the singular value decomposition (SVD)?
In the context of Principal Component Analysis (PCA), which of the following statements is/are correct?
- PCA projects the original data onto a lower-dimensional subspace thatcaptures the maximum variance.
- The eigenvectors of the covariance matrix correspond to the directions ofmaximum variance.
- The principal components form a basis that is not necessarily orthogonal.
- The principal components obtained through PCA are always uncorrelated.
Which of the following statements about singular value decomposition (SVD) is/are true?
- Singular value decomposition is possible only for a square matrix.
- Singular value decomposition is possible only for a symmetric matrix.
- The singular values of a matrix are always non-negative.
- The singular values of matrices A and AT are the same.
In the context of Linear Algebra, which of the following statements about Principal Component Analysis (PCA) is/are true?
- The principal components of a matrix are given by the eigenvectors of itscovariance matrix.
- The principal components of a matrix are given by the singular values of itscovariance matrix.
- PCA finds the principal components by maximizing the variance along eachcomponent.
- The principal components form a basis that is not necessarily orthogonal.
Comprehension passage for this question
Based on the above data, answer the given subquestions.
Compute the sample covariance matrix C for the given dataset.
Comprehension passage for this question
Based on the above data, answer the given subquestions.
What is the principal direction chosen for PCA?
Which of the following is/are true about PCA?
- PCA will transform the original data set onto a lower dimension subspace suchthat the variance of the project is maximized.
- PCA calculates the mean of each data set to determine its significance.
- PCA can be used to reduce the dimensionality of the dataset.
- PCA will transform the original data set onto a lower dimension subspace suchthat the reconstruction error is maximized.
What is the primary goal of Principal Component Analysis (PCA)?
- To find the directions that capture the maximum variance in the data.
- To find directions that capture the minimum variance in the data.
- To increase the dimensionality of the data.
- To reduce the number of data points.
Comprehension passage for this question
Based on the above data, answer the given subquestions.
Which is the principal direction that is chosen for performing PCA?
Comprehension passage for this question
Which of the following represents the covariance matrix, C for the given data points?
Which among the following statements are true about principal component analysis (PCA)?
- PCA is a supervised learning algorithm.
- Principal components are the eigenvectors of the sample covariance matrix.
- Principal components are eigenvectors of the centered data matrix.
- If p1 and p2 are the principal component vectors, then p1 is orthogonal to p2.
Which of the following statements is/are true for a real symmetric matrix A?
- AT A and AAT have the same rank.
- AT A and AAT have the same eigenvalues
- AT A and AAT have the same eigenvectors.
- AT A and AAT have the same singular values.
Which among the following is true for PCA?
- The first principal component represents the direction along which thevariance of the dataset is maximized.
- The first principal component represents the direction along which thevariance of the dataset is minimized.
- The first principal component is the eigenvector corresponding to the smallesteigenvalue of the covariance matrix.
- The first principal component is the eigenvector corresponding to the largesteigenvalue of the covariance matrix.
What are the properties of principal components in PCA?
- All principal components are orthogonal to each other.
- The number of principal components for n-dimensional data are atmost n−1.
- The first principal component accounts for most of the possible variability ofthe original data i.e, maximum possible variance.
- The first principal component is the eigenvector of the covariance matrixcorresponding to the maximum eigenvalue.