v2009.01.01 - Convex Optimization

426 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
D(X) = D(X[0 √ 2V N ]) = D(Q p [0 √ ΛQ T ]) = D([0 √ ΛQ T ])
(1039)
This suggests a way to find EDM D given −V T N DV N ; (confer (918))
[
0
D =
δ ( −VN TDV )
N
] [
1 T + 1 0 δ ( −VN TDV ) ] T
N
5.12.2 0 geometric center. V
[ 0 0
T
− 2
0 −VN TDV N
(818)
Alternatively, we may perform reconstruction using instead the auxiliary
matrix V (B.4.1), corresponding to the polyhedron
P − α c (1040)
whose geometric center has been translated to the origin. Redimensioning
diagonalization factors Q, Λ∈R N×N and unknown Q p ∈ R n×N , (941)
−V DV = 2V X T XV ∆ = Q √ ΛQ T pQ p
√
ΛQ
T ∆ = QΛQ T (1041)
where the geometrically centered generating list constitutes (confer (1037))
XV = 1 √
2
Q p
√
ΛQ T ∈ R n×N
= [x 1 − 1 N X1 x 2 − 1 N X1 x 3 − 1 N X1 · · · x N − 1 N X1 ] (1042)
where α c = 1 N X1. (5.5.1.0.1) The simplest choice for Q p is [I 0 ]∈ R r×N .
Now EDM D can be uniquely made from the list found, by calculating:
(798)
]
D(X) = D(XV ) = D( 1 √
2
Q p
√
ΛQ T ) = D( √ ΛQ T ) 1 2
(1043)
This EDM is, of course, identical to (1039). Similarly to (818), from −V DV
we can find EDM D ; (confer (905))
D = δ(−V DV 1 2 )1T + 1δ(−V DV 1 2 )T − 2(−V DV 1 2 ) (824)