v2009.01.01 - Convex Optimization

456 CHAPTER 6. CONE OF DISTANCE MATRICES
maximization:
maximize 〈−V , D〉
D
subject to 〈D , e i e T j + e j e T i 〉 1 = 2 ďij ∀(i,j)∈ I
(1091)
rank(V DV ) = 2
D ∈ EDM N
where e i ∈ R N is the i th member of the standard basis, where set I indexes
the given distance-square data like that in (1090), where V ∈ R N×N is the
geometric centering matrix (B.4.1), and where
〈−V , D〉 = tr(−V DV ) = 2 trG = 1 ∑
d ij (823)
N
where G is the Gram matrix producing D assuming G1 = 0.
If the (rank) constraint on affine dimension is ignored, then problem
(1091) becomes convex, a corresponding solution D ⋆ can be found, and
a nearest rank-2 solution is then had by ordered eigen decomposition of
−V D ⋆ V followed by spectral projection (7.1.3) on
i,j
[
R
2
+
0
]
⊂ R N .
This
two-step process is necessarily suboptimal. Yet because the decomposition
for the trefoil knot reveals only two dominant eigenvalues, the spectral
projection is nearly benign. Such a reconstruction of point position (5.12)
utilizing 4 nearest neighbors is drawn in Figure 117(b); a low-dimensional
embedding of the trefoil knot.
This problem (1091) can, of course, be written equivalently in terms of
Gram matrix G , facilitated by (829); videlicet, for Φ ij as in (796)
maximize 〈I , G〉
G∈S N c
subject to 〈G , Φ ij 〉 = ďij
rankG = 2
G ≽ 0
∀(i,j)∈ I
(1092)
The advantage to converting EDM to Gram is: Gram matrix G is a bridge
between point list X and EDM D ; constraints on any or all of these
three variables may now be introduced. (Example 5.4.2.2.4) Confinement
to the geometric center subspace S N c (implicit constraint G1 = 0) keeps G
independent of its translation-invariant subspace S N⊥
c (5.5.1.1, Figure 124)
so as not to become numerically unbounded.