I have a data matrix $M$ that has $n$ samples (rows) described by $m$ variables (columns) $X_1,X_2,\ldots X_m$. I do a SVD to reduce the $m$ dimensions to just 3 dimensions. I understand that the $x,y,z$ coordinates (i.e., the SVD values) are calculated from the eigenvectors of $MM^T$.
My question is, if I pick an arbitrary point in the SVD space (i.e. a value for SVD1, SVD2, SVD3, not necessarily in the data), is there a convenient way to translate that back to a set of the original variables (i.e., $X_1, X_2, \ldots X_m$)?