v2007.11.26 - Convex Optimization

5.13. RECONSTRUCTION EXAMPLES 389The 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 Π expressing theknown sorting action on the entries d of unknown D ∈ S N h , (64)d ∆ = 1 √2dvec D =⎡⎢⎣⎤d 12d 13d 23d 14d 24d 34⎥⎦.d N−1,N∈ R N(N−1)/2 (979)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(980)Πd ∈ K M+D ∈ EDM Nthat finds the EDM D (corresponding to affine dimension not exceeding 3 inisomorphic dvec EDM N ∩ Π T K M+ ) closest to O in the sense of Schoenberg(753).Analytical solution to this problem, ignoring the sort constraintΠd ∈ K M+ , is known [266]: we get the convex optimization [sic] (7.1)minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3(981)D ∈ EDM NOnly the three largest nonnegative eigenvalues in (978) need be retainedto make list (965); the rest are discarded. The reconstruction fromEDM D found in this manner is plotted in Figure 96(e)(f) from which itbecomes obvious that inclusion of the sort constraint is necessary for isotonicreconstruction.