v2006.03.09 - Convex Optimization

4.13. RECONSTRUCTION EXAMPLES 293The extra eigenvalues indicate that affine dimension corresponding to anEDM near O is likely to exceed 3. To realize the map, we must simultaneouslyreduce that dimensionality and find an EDM D closest to O in some sense(a problem explored more in7) while maintaining the known comparativedistance relationship; e.g., given permutation matrix Ξ expressing the knownsorting action on the entries d of unknown D ∈ S N h , (62)d ∆ = 1 √2dvec D =⎡⎢⎣⎤d 12d 13d 23d 14d 24d 34⎥⎦.d N−1,N∈ R N(N−1)/2 (693)we can make the sort-index matrix O input to the optimization problemminimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3(694)Ξd ∈ K M+D ∈ EDM Nthat finds the EDM D (corresponding to affine dimension not exceeding 3 inisomorphic dvec EDM N ∩ Ξ T K M+ ) closest to O in the sense of Schoenberg(479).Analytical solution to this problem, ignoring the sort constraintΞd ∈ K M+ , is known [229]: we get the convex optimization [sic] (7.1)minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3(695)D ∈ EDM NOnly the three largest nonnegative eigenvalues in (692) need be retainedto make list (679); the rest are discarded. The reconstruction fromEDM D found in this manner is plotted in Figure 70(e)(f) from which itbecomes obvious that inclusion of the sort constraint is necessary for isotonicreconstruction.