v2009.01.01 - Convex Optimization

380 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
Θ = [x 2 − x 1 x 3 − x 1 · · · x N − x 1 ] = X √ 2V N ∈ R n×N−1 (865)
Inner product Θ T Θ is obviously related to a Gram matrix (807),
[ 0 0
T
G =
0 Θ T Θ
]
, x 1 = 0 (866)
For D = D(Θ) and no condition on the list X (confer (815) (820))
Θ T Θ = −V T NDV N ∈ R N−1×N−1 (867)
5.4.3.1 Relative-angle form
The inner-product form EDM definition is not a unique definition of
Euclidean distance matrix; there are approximately five flavors distinguished
by their argument to operator D . Here is another one:
Like D(X) (798), D(Θ) will make an EDM given any Θ∈ R n×N−1 , it is
neither a convex function of Θ (5.4.3.2), and it is homogeneous in the sense
(801). Scrutinizing Θ T Θ (862) we find that because of the arbitrary choice
k = 1, distances therein are all with respect to point x 1 . Similarly, relative
angles in Θ T Θ are between all vector pairs having vertex x 1 . Yet picking
arbitrary θ i1j to fill Θ T Θ will not necessarily make an EDM; inner product
(862) must be positive semidefinite.
Θ T Θ = √ δ(d) Ω √ δ(d) =
∆
⎡√ ⎤⎡
⎤⎡√ ⎤
d12 0 1 cos θ 213 · · · cos θ 21N d12 0
√ d13 ⎢
⎥
cosθ 213 1
... cos θ √
31N
d13 ⎣ ... ⎦⎢
⎥⎢
⎥
√ ⎣ .
...
... . ⎦⎣
... ⎦
√
0
d1N cos θ 21N cos θ 31N · · · 1 0
d1N
(868)
Expression D(Θ) defines an EDM for any positive semidefinite relative-angle
matrix
Ω = [cos θ i1j , i,j = 2...N] ∈ S N−1 (869)
and any nonnegative distance vector
d = [d 1j , j = 2...N] = δ(Θ T Θ) ∈ R N−1 (870)