v2009.01.01 - Convex Optimization

5.10. EDM-ENTRY COMPOSITION 415
Then d embeds in L 2 iff p is a positive semidefinite matrix iff d
is of negative type (second half page 525/top of page 526 in [271]).
For the implication from (ii) to (iii), set: p = e −αd and define d ′
from p using (B) above. Then d ′ is a distance space on N+1
points that embeds in L 2 . Thus its subspace of N points also
embeds in L 2 and is precisely 1 − e −αd .
Note that (iii) ⇒ (ii) cannot be read immediately from this argument
since (iii) involves the subdistance of d ′ on N points (and not the full
d ′ on N+1 points).
3) Show (iii) ⇒ (i) by using the series expansion of the function 1 − e −αd :
the constant term cancels, α factors out; there remains a summation
of d plus a multiple of α . Letting α go to 0 gives the result.
This is not explicitly written in Schoenberg, but he also uses such
an argument; expansion of the exponential function then α → 0 (first
proof on [271, p.526]).
Schoenberg’s results [271,6, thm.5] (confer [197, p.108-109]) also
suggest certain finite positive roots of EDM entries produce EDMs;
specifically,
D ∈ EDM N ⇔ [d 1/α
ij ] ∈ EDM N ∀α > 1 (998)
The special case α = 2 is of interest because it corresponds to absolute
distance; e.g.,
D ∈ EDM N ⇒ ◦√ D ∈ EDM N (999)
Assuming that points constituting a corresponding list X are distinct
(960), then it follows: for D ∈ S N h
lim
α→∞ [d1/α ij
] = lim
α→∞
[1 − e −αd ij
] = −E ∆ = 11 T − I (1000)
Negative elementary matrix −E (B.3) is relatively interior to the EDM cone
(6.6) and terminal to the respective trajectories (997) and (998) as functions
of α . Both trajectories are confined to the EDM cone; in engineering terms,
the EDM cone is an invariant set [268] to either trajectory. Further, if D is
not an EDM but for some particular α p it becomes an EDM, then for all
greater values of α it remains an EDM.