v2009.01.01 - Convex Optimization

5.13. RECONSTRUCTION EXAMPLES 431
The extra eigenvalues indicate that affine dimension corresponding to an
EDM near O is likely to exceed 3. To realize the map, we must simultaneously
reduce that dimensionality and find an EDM D closest to O in some sense
(a problem explored more in7) while maintaining the known comparative
distance relationship; e.g., given permutation matrix Π expressing the
known sorting action on the entries d of unknown D ∈ S N h , (66)
d ∆ = 1 √
2
dvec D =
⎡
⎢
⎣
⎤
d 12
d 13
d 23
d 14
d 24
d 34
⎥
⎦
.
d N−1,N
∈ R N(N−1)/2 (1051)
we can make the sort-index matrix O input to the optimization problem
minimize ‖−VN T(D − O)V N ‖ F
D
subject to rankVN TDV N ≤ 3
(1052)
Πd ∈ K M+
D ∈ EDM N
that finds the EDM D (corresponding to affine dimension not exceeding 3 in
isomorphic dvec EDM N ∩ Π T K M+ ) closest to O in the sense of Schoenberg
(817).
Analytical solution to this problem, ignoring the sort constraint
Πd ∈ K M+ , is known [306]: we get the convex optimization [sic] (7.1)
minimize ‖−VN T(D − O)V N ‖ F
D
subject to rankVN TDV N ≤ 3
(1053)
D ∈ EDM N
Only the three largest nonnegative eigenvalues in (1050) need be retained to
make list (1037); the rest are discarded. The reconstruction from EDM D
found in this manner is plotted in Figure 110(e)(f). Matlab code is on
Wıκımization. From these plots it becomes obvious that inclusion of the
sort constraint is necessary for isotonic reconstruction.