Question 10: Consider a dataset of 20 points where the i^{th} data-po…

Consider a dataset of 20 points where the $i^{th}$ data-point is given by:

$\mathbf{x}_i = a_i \cdot \begin{bmatrix} 1 \\ 1 \\ 0 \\ 2 \end{bmatrix} + b_i \cdot \begin{bmatrix} -1 \\ 1 \\ 3 \\ 0 \end{bmatrix}$

where, $a_i$ and $b_i$ are real numbers such that $\sum_{i=1}^{20} a_i = \sum_{i=1}^{20} b_i = 0$. Standard PCA is performed on this dataset. If the top two principal components are retained and used to reconstruct the dataset, what is the reconstruction error?

Hint: Think about what happens in $\mathbb{R}^2$ or $\mathbb{R}^3$ for a similar situation and extend this idea to $\mathbb{R}^4$.