v2010.10.26 - Convex Optimization

5.10. EDM-ENTRY COMPOSITION 467Then d embeds in L 2 iff p is a positive semidefinite matrix iff dis of negative type (second half page 525/top of page 526 in [313]).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+1points that embeds in L 2 . Thus its subspace of N points alsoembeds in L 2 and is precisely 1 − e −αd .Note that (iii) ⇒ (ii) cannot be read immediately from this argumentsince (iii) involves the subdistance of d ′ on N points (and not the fulld ′ 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 summationof 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 suchan argument; expansion of the exponential function then α → 0 (firstproof on [313, p.526]).Schoenberg’s results [313,6 thm.5] (confer [227, p.108-109]) also suggestcertain finite positive roots of EDM entries produce EDMs; specifically,D ∈ EDM N ⇔ [d 1/αij ] ∈ EDM N ∀α > 1 (1093)The special case α = 2 is of interest because it corresponds to absolutedistance; e.g.,D ∈ EDM N ⇒ ◦√ D ∈ EDM N (1094)Assuming that points constituting a corresponding list X are distinct(1055), then it follows: for D ∈ S N hlimα→∞ [d1/α ij] = limα→∞[1 − e −αd ij] = −E 11 T − I (1095)Negative elementary matrix −E (B.3) is relatively interior to the EDM cone(6.5) and terminal to respective trajectories (1092a) and (1093) as functionsof α . Both trajectories are confined to the EDM cone; in engineering terms,the EDM cone is an invariant set [310] to either trajectory. Further, if D isnot an EDM but for some particular α p it becomes an EDM, then for allgreater values of α it remains an EDM.