MMiDS 4.4: Self-Assessment Quiz

In the power iteration lemma for the positive semidefinite case, what happens when the initial vector $x$ satisfies $⟨q_1,x⟩<0$?

• 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) $B=A{A}^T$
• 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) $(A^{T}A{)}^k\approx {\sigma }_1^{2k}{v}_1{v}_1^T$
• d) $(A^{T}A{)}^k\approx {\sigma }_2^{2k}{v}_2{v}_2^T$

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

• a) $B^{k}x/\Vert B^{k}x\Vert$ converges to $u_1$.
• b) $B^{k}x/\Vert B^{k}x\Vert$ converges to $v_1$ or $-v_1$.
• c) $B^{k}x/\Vert B^{k}x\Vert$ converges to ${\sigma }_1$.
• d) $B^{k}x/\Vert B^{k}x\Vert$ 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) $\sigma =\Vert Av\Vert$ and $u=Av/\sigma$
• b) $\sigma =\Vert A^{T}v\Vert$ and $u=A^{T}v/\sigma$
• c) $\sigma =\Vert v\Vert$ 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) $⟨q_1,x⟩\ne 0$.
• 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 $⟨v_1,X⟩=0$?

• 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 $Z=U_{(2)}{\mathrm{\Sigma }}_{(2)}{V}_{(2)}^T$ 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