MMiDS 4.4: Self-Assessment Quiz

In the power iteration lemma for the positive semidefinite case, what happens when the initial vector \( x \) satisfies \( \langle q_1, x \rangle < 0 \)?

• a) The iteration converges to \( q_1 \).
• b) The iteration converges to \( -q_1 \).
• c) The iteration does not converge.
• d) The iteration converges to a random eigenvector.

What is the relationship between the matrix \( B \) and the matrix \( A \) in the power iteration lemma for the SVD case?

• a) \( B = A \)
• b) \( B = A^T \)
• c) \( B = AA^T \)
• d) \( B = A^T A \)

If \( \sigma_1 > \sigma_2 > 0 \) are the top two singular values of a matrix \( A \), and \( k \) is a large positive integer, which of the following approximations is used in the power iteration method?

• a) \( A^k \approx \sigma_1^k u_1 v_1^T \)
• b) \( A^k \approx \sigma_2^k u_2 v_2^T \)
• c) \( (A^TA)^k \approx \sigma_1^{2k} v_1 v_1^T \)
• d) \( (A^TA)^k \approx \sigma_2^{2k} v_2 v_2^T \)

In the power iteration lemma for the SVD case, what is the convergence result for a random vector \( x \)?

• a) \( B^k x / \|B^k x\| \) converges to \( u_1 \).
• b) \( B^k x / \|B^k x\| \) converges to \( v_1 \) or \( -v_1 \).
• c) \( B^k x / \|B^k x\| \) converges to \( \sigma_1 \).
• d) \( B^k x / \|B^k x\| \) does not converge.

Suppose you apply the power iteration method to a matrix \( A \) and obtain a vector \( v \). How can you compute the corresponding singular value \( \sigma \) and left singular vector \( u \)?

• a) \( \sigma = \|Av\| \) and \( u = Av/\sigma \)
• b) \( \sigma = \|A^Tv\| \) and \( u = A^Tv/\sigma \)
• c) \( \sigma = \|v\| \) and \( u = v/\sigma \)
• d) \( \sigma = 1 \) and \( u = Av \)

What is required for the initial vector \( x \) in the power iteration method to ensure convergence to the top eigenvector?

• a) \( x \) must be a zero vector.
• b) \( x \) must be orthogonal to the top eigenvector.
• c) \( \langle q_1, x \rangle \neq 0 \).
• d) \( x \) must be the top eigenvector itself.

What is the probability that a random \( m \)-dimensional spherical Gaussian vector \( X \) with mean \( 0 \) and variance \( 1 \) satisfies \( \langle v_1, X \rangle = 0 \)?

• a) 0
• b) 1
• c) 0.5
• d) \( 1/m \)

In the orthogonal iteration method for computing multiple singular vectors, what is done after each application of \( B \) to the subspace?

• a) The resulting vectors are normalized.
• b) An orthonormal basis is calculated for the resulting subspace.
• c) The resulting vectors are projected onto the top singular vector.
• d) The resulting vectors are discarded.

What is the main advantage of projecting data points onto the top right singular vectors before clustering?

• a) It reduces the dimensionality of the data.
• b) It increases the separation between clusters.
• c) It brings data points closer to their cluster centers.
• d) It removes outliers from the data.

What does the truncated SVD \( Z = U_{(2)} \Sigma_{(2)} V_{(2)}^T \) correspond to?

• a) The best one-dimensional approximating subspace
• b) The best two-dimensional approximating subspace
• c) The projection of the data onto the top singular vector
• d) The projection of the data onto the top two singular vectors