Quizzes: SVD for dimension reduction

by Kurious Fox

Where does SVD shine for dimensionality reduction?

A) Image compression ??
B) Text mining ?
C) Recommendation systems ?
D) All of the above ?

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Answer: D) All of the above
Explanation: SVD is used in various fields such as image compression, text mining (e.g., Latent Semantic Analysis), and recommendation systems for reducing dimensionality while retaining key information.

How does Singular Value Decomposition (SVD) come to the rescue for noise reduction?

A) By cutting out features with low variance ??
B) By zeroing out small singular values, removing less significant components ?
C) By giving the data a fresh coat (standardizing) ??
D) By grouping similar data points into clusters ?

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Answer: B) By setting small singular values to zero, effectively removing less significant components
Explanation: By truncating small singular values, SVD removes less significant components that often correspond to noise, thus improving the signal-to-noise ratio.

Which of the following matrices are obtained from performing SVD on a matrix A ?
A) Covariance matrix and eigenvectors
B) Orthogonal matrices U and V , and diagonal matrix \Sigma
C) Mean and standard deviation matrices
D) Identity matrices

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Answer: B) Orthogonal matrices U and V , and diagonal matrix \Sigma
Explanation: SVD decomposes a matrix A into three matrices: U (left singular vectors), \Sigma (singular values), and V (right singular vectors).

What do the singular values in the diagonal matrix \Sigma represent?
A) The variance explained by each principal component
B) The correlation between the features
C) The scaling factors for the transformation
D) The square roots of the eigenvalues of A^T A

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Answer: D) The square roots of the eigenvalues of A^T A
Explanation: The singular values in \Sigma are the square roots of the eigenvalues of A^T A , and they indicate the magnitude of each component in the decomposition.

True or False: SVD can only be applied to square matrices.
A) True
B) False

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Answer: B) False
Explanation: SVD can be applied to any m \times n matrix, not just square matrices.

Which equation correctly represents the Singular Value Decomposition of a matrix A ?
A) A = U \Sigma V^T
B) A = UV \Sigma
C) A = \Sigma U V^T
D) A = U V \Sigma^T

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Answer: A) A = U \Sigma V^T
Explanation: The correct representation of SVD is A = U \Sigma V^T , where A is decomposed into an orthogonal matrix U , a diagonal matrix \Sigma , and the transpose of an orthogonal matrix V .

In the context of SVD, what are the matrices U and V ?
A) Diagonal matrices
B) Orthogonal matrices
C) Identity matrices
D) Covariance matrices

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Answer: B) Orthogonal matrices
Explanation: U and V are orthogonal matrices containing the left and right singular vectors, respectively.

What property do the columns of the orthogonal matrices U and V have?
A) They are orthonormal
B) They are linearly dependent
C) They are eigenvectors of A
D) They are random vectors

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Answer: A) They are orthonormal
Explanation: The columns of U and V are orthonormal, meaning they are orthogonal (perpendicular) and each column vector has a unit norm (length of 1).

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