If the objective is to preserve these distances, using only d dimensions, classical MDS theory says to use the N × d-dimensional matrix with row i (2)(λ11/2u1(i),…,λd1/2ud(i)) representing the ith individual. Let m̂ (i, j) be the Euclidean distance between individuals i and j in this low-dimensional configuration. To measure the discrepancy between the Euclidean distances in the full and low-dimensional space, let δ = ∑i,j (m(i, j)2 − m̂(i, j)2). This quantity is minimized over all d-dimensional configurations by the top d eigenvectors of H, weighted by the square root of the eigenvalues (Equation 2) [Mardia et al., 1979]. Thus PC mapping is a type of classical MDS. It provides the optimal embedding if the goal is to preserve the pairwise distances m(i,j) with H = L−1ϒϒt as closely as possible.
