I have a collection of vectors (a codebook in hand) which are presented within a matrix $A$
$$ A = [ a (\theta_1), \, a (\theta_2), \, a (\theta_3), \, a (\theta_4), \,\cdots]$$
We have obtained from a certain observation some vector, lets call it $a(\phi)$, which is not from any of the vectors in matrix $A$, our aim however is to find a vector from A that is closest and best resembles $a(\phi)$
I have thought of couple of solutions
1-$i^*$= $\arg \min_i$ $|| a(\theta_i) - a (\phi) ||^2$
2-$i^*$= $\arg \min_i \mathbb{E} [ |e_i|^2]$ where $e_i=a(\theta_i) - a (\phi) $ (minimum mean square error estimate)
3- $i^* = \arg \max_i |a(\theta_i)a (\phi)|$
Which of these solutions is the best, are they all the same? Any reference on how to handle such problems?