v2004.06.19 - Convex Optimization

176 CHAPTER 4. EUCLIDEAN DISTANCE MATRIXbut the comparative data is available:d 12 ≤ d 23 ≤ d 13 (499)With the vectorization d = [d 12 d 13 d 23 ] T ∈ R 3 , we express the comparativedistance relationship as the nondecreasing sorting⎡ ⎤⎡⎤ ⎡ ⎤1 0 0 d 12 d 12Ξd = ⎣ 0 0 1 ⎦⎣d 13⎦ = ⎣ d 23⎦ ∈ K +M (500)0 1 0 d 23 d 13where Ξ is a given permutation matrix expressing the known sorting actionon the entries of unknown EDM D , and K +M is the monotone nonnegativecone (§2.9.2.2.1) with reversed indices,K +M ∆ = {z | 0 ≤ z 1 ≤ z 2 ≤ · · · ≤ z N(N−1)/2 } ⊆ R N(N−1)/2+ (501)where N(N − 1)/2 = 3 for the present example. From the sorted vectorization(500) we create the sort-index matrix⎡ ⎤0 1 2 3 2O = ⎣ 1 2 0 2 2 ⎦ ∈ S 33 2 2 2 0 ∩ R 3×3+ (502)0where, for j ≠ i ,O ij = k 2 | d ij = ( Ξd ) k(503)Replacing EDM data with indices-square of a nondecreasing sorting likethis is, of course, a heuristic we invented. Any process of reconstructionthat leaves comparative distance information intact is called ordinalmultidimensional scaling or isotonic reconstruction. Beyond rotation, reflection,and translation error, (§4.5) list reconstruction by isotonic reconstructionis subject to error in absolute scale (dilation) and distance ratio. YetBorg and Groenen argue: [100, §2.2] reconstruction from complete comparativedistance information for a large number of points is as highly constrainedas reconstruction from an EDM; the larger the number, the better.4.11.2.0.1 Example. Isotonic map of the USA.To test that conjecture, suppose we make a complete sort-index matrixO ∈ S N 0 ∩ R N×N+ for the map of the USA and then substitute O in place of