v2010.10.26 - Convex Optimization

5.13. RECONSTRUCTION EXAMPLES 481With vectorization d = [d 12 d 13 d 23 ] T ∈ R 3 , we express the comparative dataas the nonincreasing sorting⎡ ⎤⎡⎤ ⎡ ⎤0 1 0 d 12 d 13Πd = ⎣ 0 0 1 ⎦⎣d 13⎦ = ⎣ d 23⎦ ∈ K M+ (1142)1 0 0 d 23 d 12where Π is a given permutation matrix expressing known sorting action onthe entries of unknown EDM D , and K M+ is the monotone nonnegativecone (2.13.9.4.2)K M+ = {z | z 1 ≥ z 2 ≥ · · · ≥ z N(N−1)/2 ≥ 0} ⊆ R N(N−1)/2+ (429)where N(N −1)/2 = 3 for the present example. From sorted vectorization(1142) we create the sort-index matrixgenerally defined⎡O = ⎣0 1 2 3 21 2 0 2 23 2 2 2 0⎤⎦ ∈ S 3 h ∩ R 3×3+ (1143)O ij k 2 | d ij = (Ξ Πd) k, j ≠ i (1144)where Ξ is a permutation matrix (1728) completely reversing order of vectorentries.Replacing EDM data with indices-square of a nonincreasing sorting likethis is, of course, a heuristic we invented and may be regarded as a nonlinearintroduction of much noise into the Euclidean distance matrix. For largedata sets, this heuristic makes an otherwise intense problem computationallytractable; we see an example in relaxed problem (1148).Any process of reconstruction that leaves comparative distanceinformation intact is called ordinal multidimensional scaling or isotonicreconstruction. Beyond rotation, reflection, and translation error, (5.5)list reconstruction by isotonic reconstruction is subject to error in absolutescale (dilation) and distance ratio. Yet Borg & Groenen argue: [53,2.2]reconstruction from complete comparative distance information for a largenumber of points is as highly constrained as reconstruction from an EDM;the larger the number, the better.