Given two vector sets, $\vec x_i$ and $\vec y_i$ (for $i$=1,2,...N, but the dimensionality of each vector can be more than N), let their sum set be $\vec z_i = \vec x_i + \vec y_i$. It's easy to compute the scalar inter-distances: $x_{ij} = \Vert{\vec x_i-\vec x_j}\Vert$, $y_{ij} = \Vert{\vec y_i-\vec y_j}\Vert$, $z_{ij} = \Vert{\vec z_i-\vec z_j}\Vert$.

But, what's the quickest computation method to go backwards (i.e., from all of the inter-distances to any "high-dimensional embedding")? It seems I'm forced into ugly numerical "trial and error" methods even for small N=4 though there are easy analytical methods (for any N) if I didn't need to match the sum's inter-distances.