\begin{problem}{Conceptual questions}

\begin{question} Suppose we have three data points in 2-D space, (1, 1), (2, 2) and (3, 3).

\begin{subquestion} [2] What is the first principal component?

\solutionspace{1cm}{14cm}{}{\OneAi}

\end{subquestion}

\begin{subquestion} [2] If we want to project the original data points into 1-D space using this principal component, what will be the variance of the projected data?

\solutionspace{1cm}{14cm}{}{\OneAii}

\end{subquestion}

\begin{subquestion} [2] Suppose we want to reconstruct the original 2-D data using our projected data from (b). What is the reconstruction error in this case? Justify your answer.

\solutionspace{2cm}{14cm}{}{\OneAiii}

\end{subquestion}

\end{question}

\begin{question}[2] 
True or False:
It is reasonable to pick the number of clusters K in K-means clustering by minimizing the sum of squared distances between the data points and the cluster centers. Explain your answer.

\solutionspace{3cm}{15cm}{}{\OneB}

\end{question}


\begin{question}[2] Recall that in the E-step of the EM algorithm, we ``softly'' assign each data point to the the clusters. Under what parameter setting will the soft assignment in the EM algorithm with a univariate GMM reduce to hard assignment (as we've discussed in K-means)?

\solutionspace{3cm}{15cm}{}{\OneC}

\end{question}

\begin{question}[2] True or False: The EM algorithm will always converge to the global minimum because each EM step will monotonically improve the likelihood. Explain your answer.

\solutionspace{3cm}{15cm}{}{\OneD}

\end{question}
\end{problem}





