v2010.10.26 - Convex Optimization

5.13. RECONSTRUCTION EXAMPLES 483d 1 √2dvec D =⎡⎢⎣⎤d 12d 13d 23d 14d 24d 34⎥⎦.d N−1,N∈ R N(N−1)/2 (1146)of unknown D ∈ S N h , we can make sort-index matrix O input to theoptimization problemminimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3(1147)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(910).Analytical solution to this problem, ignoring the sort constraintΠd ∈ K M+ , is known [353]: we get the convex optimization [sic] (7.1)minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3(1148)D ∈ EDM NOnly the three largest nonnegative eigenvalues in (1145) need be retained tomake list (1132); the rest are discarded. The reconstruction from EDM Dfound in this manner is plotted in Figure 134e-f. Matlab code is onWıκımization. From these plots it becomes obvious that inclusion of thesort constraint is necessary for isotonic reconstruction.That sort constraint demands: any optimal solution D ⋆ must possess theknown comparative distance relationship that produces the original ordinaldistance data O (1144). Ignoring the sort constraint, apparently, violates it.Yet even more remarkable is how much the map reconstructed using onlyordinal data still resembles the original map of the USA after suffering themany violations produced by solving relaxed problem (1148). This suggeststhe simple reconstruction techniques of5.12 are robust to a significantamount of noise.