v2007.09.13 - Convex Optimization

5.10. EDM-ENTRY COMPOSITION 357(ii) ⇒ (iii): set p = e −αd and define d ′ from p using (B) above. Thend ′ is a distance space on N+1 points that embeds in L 2 . Thus itssubspace of N points also embeds 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 [233, p.526]).Schoenberg’s results [233,6, thm.5] (confer [165, p.108-109]) alsosuggest certain finite positive roots of EDM entries produce EDMs;specifically,D ∈ EDM N ⇔ [d 1/αij ] ∈ EDM N ∀α > 1 (896)The special case α = 2 is of interest because it corresponds to absolutedistance; e.g.,D ∈ EDM N ⇒ ◦√ D ∈ EDM N (897)Assuming that points constituting a corresponding list X are distinct(860), then it follows: for D ∈ S N hlimα→∞ [d1/α ij] = limα→∞[1 − e −αd ij] = −E ∆ = 11 T − I (898)Negative elementary matrix −E (B.3) is relatively interior to the EDM cone(6.6) and terminal to the respective trajectories (895) and (896) as functionsof α . Both trajectories are confined to the EDM cone; in engineering terms,the EDM cone is an invariant set [230] 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.