Given $\{$xi$\}$

N

i$=1,$ N data points each of d dimensions, you run Principal Component Analysis

and pick P principal components. Can you always reconstruct any data point xi

for i in $\{1$ N$\}$

from the P principal components with zero reconstruction error?

$($a$)$ Yes, always $($b$)$ Yes, if P $<$ d $($c$)$ Yes, if P $<$ N $($d$)$ Yes, if P $=$ d