v2009.01.01 - Convex Optimization

432 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
That sort constraint demands: any optimal solution D ⋆ must possess the
known comparative distance relationship that produces the original ordinal
distance data O (1049). Ignoring the sort constraint, apparently, violates it.
Yet even more remarkable is how much the map reconstructed using only
ordinal data still resembles the original map of the USA after suffering the
many violations produced by solving relaxed problem (1053). This suggests
the simple reconstruction techniques of5.12 are robust to a significant
amount of noise.
5.13.2.2 Isotonic solution with sort constraint
Because problems involving rank are generally difficult, we will partition
(1052) into two problems we know how to solve and then alternate their
solution until convergence:
minimize ‖−VN T(D − O)V N ‖ F
D
subject to rankVN TDV N ≤ 3
D ∈ EDM N
minimize ‖σ − Πd‖
σ
subject to σ ∈ K M+
(a)
(b)
(1054)
where the sort-index matrix O (a given constant in (a)) becomes an implicit
vectorized variable o i solving the i th instance of (1054b)
o i ∆ = Π T σ ⋆ = 1 √
2
dvec O i ∈ R N(N−1)/2 , i∈{1, 2, 3...} (1055)
As mentioned in discussion of relaxed problem (1053), a closed-form
solution to problem (1054a) exists. Only the first iteration of (1054a) sees the
original sort-index matrix O whose entries are nonnegative whole numbers;
id est, O 0 =O∈ S N h ∩R N×N
+ (1049). Subsequent iterations i take the previous
solution of (1054b) as input
O i = dvec −1 ( √ 2o i ) ∈ S N (1056)
real successors to the sort-index matrix O .
New problem (1054b) finds the unique minimum-distance projection of
Πd on the monotone nonnegative cone K M+ . By defining
Y †T ∆ = [e 1 − e 2 e 2 −e 3 e 3 −e 4 · · · e m ] ∈ R m×m (384)