v2009.01.01 - Convex Optimization

5.13. RECONSTRUCTION EXAMPLES 429
D = [d ij ] =
⎡
⎣
0 d 12 d 13
d 12 0 d 23
d 13 d 23 0
but the comparative data is available:
⎤
⎦ ∈ S 3 h (787)
d 13 ≥ d 23 ≥ d 12 (1046)
With the vectorization d = [d 12 d 13 d 23 ] T ∈ R 3 , we express the comparative
distance relationship as the nonincreasing sorting
Πd =
⎡
⎣
0 1 0
0 0 1
1 0 0
⎤⎡
⎦⎣
⎤
d 12
d 13
⎦ =
d 23
⎡
⎣
⎤
d 13
d 23
⎦ ∈ K M+ (1047)
d 12
where Π is a given permutation matrix expressing the known sorting action
on the entries of unknown EDM D , and K M+ is the monotone nonnegative
cone (2.13.9.4.2)
K M+ ∆ = {z | z 1 ≥ z 2 ≥ · · · ≥ z N(N−1)/2 ≥ 0} ⊆ R N(N−1)/2
+ (383)
where N(N −1)/2 = 3 for the present example.
vectorization (1047) we create the sort-index matrix
generally defined
⎡
O = ⎣
0 1 2 3 2
1 2 0 2 2
3 2 2 2 0
⎤
From the sorted
⎦ ∈ S 3 h ∩ R 3×3
+ (1048)
O ij ∆ = k 2 | d ij = ( Ξ Πd ) k , j ≠ i (1049)
where Ξ is a permutation matrix (1608) completely reversing order of vector
entries.
Replacing EDM data with indices-square of a nonincreasing sorting like
this is, of course, a heuristic we invented and may be regarded as a nonlinear
introduction of much noise into the Euclidean distance matrix. For large
data sets, this heuristic makes an otherwise intense problem computationally
tractable; we see an example in relaxed problem (1053).