v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 379
where ([√the numerator ]) forms an inner product of vectors. Distance-square
dik
d ij √dkj is a convex quadratic function 5.20 on R 2 + whereas d ij (θ ikj ) is
quasiconvex (3.3) minimized over domain {−π ≤θ ikj ≤π} by θ ⋆ ikj =0, we
get the Pythagorean theorem when θ ikj = ±π/2, and d ij (θ ikj ) is maximized
when θ ⋆ ikj =±π ; d ij = (√ d ik + √ ) 2,
d kj θikj = ±π
so
d ij = d ik + d kj , θ ikj = ± π 2
d ij = (√ d ik − √ d kj
) 2,
θikj = 0
(860)
| √ d ik − √ d kj | ≤ √ d ij ≤ √ d ik + √ d kj (861)
Hence the triangle inequality, Euclidean metric property 4, holds for any
EDM D .
We may construct an inner-product form of the EDM definition for
matrices by evaluating (858) for k=1: By defining
⎡
√ √ √ ⎤
d 12 d12 d 13 cosθ 213 d12 d 14 cos θ 214 · · · d12 d 1N cos θ 21N
√ √ √ d12 d
Θ T Θ =
∆ 13 cos θ 213 d 13 d13 d 14 cos θ 314 · · · d13 d 1N cos θ 31N
√ √ √ d12 d 14 cos θ 214 d13 d 14 cosθ 314 d 14
... d14 d 1N cos θ 41N
∈ S N−1
⎢
⎥
⎣ .
.
...
...
√ √
. ⎦
√
d12 d 1N cos θ 21N d13 d 1N cosθ 31N d14 d 1N cos θ 41N · · · d 1N
then any EDM may be expressed
[ ]
D(Θ) =
∆ 0
δ(Θ T 1
Θ)
T + 1 [ [
0 δ(Θ T Θ) T] 0 0
T
− 2
0 Θ T Θ
=
[
0 δ(Θ T Θ) T
δ(Θ T Θ) δ(Θ T Θ)1 T + 1δ(Θ T Θ) T − 2Θ T Θ
]
]
(862)
∈ EDM N
(863)
EDM N = { D(Θ) | Θ ∈ R N−1×N−1} (864)
for which all Euclidean metric properties hold. Entries of Θ T Θ result from
vector inner-products as in (859); id est,
[
]
5.20 1 −e ıθ ikj
−e −ıθ ≽ 0, having eigenvalues {0,2}. Minimum is attained for
ikj
1
[ √ ] {
dik
√dkj
µ1, µ ≥ 0, θikj = 0
=
0, −π ≤ θ ikj ≤ π , θ ikj ≠ 0 . (D.2.1, [53, exmp.4.5])