v2006.03.09 - Convex Optimization

4.13. RECONSTRUCTION EXAMPLES 291D = [d ij ] =⎡⎣0 d 12 d 13d 12 0 d 23d 13 d 23 0but the comparative data is available:⎤⎦ ∈ S 3 h (449)d 13 ≥ d 23 ≥ d 12 (688)With the vectorization d = [d 12 d 13 d 23 ] T ∈ R 3 , we express the comparativedistance relationship as the nonincreasing sortingΞd =⎡⎣0 1 00 0 11 0 0⎤⎡⎦⎣⎤d 12d 13⎦ =d 23⎡⎣⎤d 13d 23⎦ ∈ K M+ (689)d 12where Ξ is a given permutation matrix expressing the known sorting actionon the entries of unknown EDM D , and K M+ is the monotone nonnegativecone (2.13.9.4.1)K M+ ∆ = {z | z 1 ≥ z 2 ≥ · · · ≥ z N(N−1)/2 ≥ 0} ⊆ R N(N−1)/2+ (347)where N(N −1)/2 = 3 for the present example.vectorization (689) we create the sort-index matrixgenerally defined⎡O = ⎣0 1 2 3 21 2 0 2 23 2 2 2 0⎤From the sorted⎦ ∈ S 3 h ∩ R 3×3+ (690)O ij ∆ = k 2 | d ij = ( Π Ξd ) k , j ≠ i (691)where Π is a permutation matrix completely reversing order of vector entries.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 (695).