Algorithm 3.

(MDS) Given weights W and distances Y ∈ ℝ^q×q, find the matrix Θ = [θ¹, …, θ^q] ∈ ℝ^p×q which minimizes the stress (3.8).

Precompute: xij ← wij yij for all 1 ≤ i, j ≤ q

Precompute: wi· ← Σj wij for all 1 ≤ i ≤ q

Initialize: Draw ${\theta }_{0k}^i$ uniformly on [−1,1] for all 1 ≤ i ≤ q, 1 ≤ k ≤ p

repeat

Compute ${\mathbf{\Theta }}_n^t{\mathbf{\Theta }}_n$

$d_{\mathit{nij}}\leftarrow {\{{\mathbf{\Theta }}_n^t{\mathbf{\Theta }}_n\}}_{ii}+{\{{\mathbf{\Theta }}_n^t{\mathbf{\Theta }}_n\}}_{jj}-2{\{{\mathbf{\Theta }}_n^t{\mathbf{\Theta }}_n\}}_{ij}$ for all 1 ≤ i, j ≤ q

znij ← xij/dnij for all 1 ≤ i ≠ j ≤ q

zni· ← Σj znij for all 1 ≤ i ≤ q

Compute Θ;n(W − Zn)

${\theta }_{n+1,k}^i\leftarrow [{\theta }_{nk}^i(w_{i·}+z_{ni·})+{\{{\mathbf{\Theta }}_n(\mathbf{W}-{\mathbf{Z}}_n)\}}_{ik}]/(2w_{i·})$ for all 1 ≤ i ≤ p, 1 ≤ k ≤ q

until convergence occurs