I have a set of data that has $n$ samples described by $m$ variables. I do a PCA to reduce it to just 2 dimensions so I can make a nice 2D plot of the data. I understand that the $x,y$ coordinates (i.e., the PCA scores) for the plot are calculated by basically summing the products of the original data (after centering) by the loadings for each variable, so:
$PC_1 = X_1L_1 + X_2L_2 + ... + X_nL_n$
My question is, if I pick an arbitrary point in the PCA space (i.e. a value for $PC_1$ and $PC_2$, or $x$ and $y$ in my plot), is there a convenient way to translate that back to a set of the original values (i.e., $X_1,X_2,\dots,X_n$)?
Note 100% reversal is obviously not expected (since I'm only using 2 PCs), so a decent approximation is fine.