v2007.09.13 - Convex Optimization

356 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.10 EDM-entry compositionLaurent [170,2.3] applies results from Schoenberg (1938) [233] to showcertain nonlinear compositions of individual EDM entries yield EDMs;in particular,D ∈ EDM N ⇔ [1 − e −αd ij] ∈ EDM N ∀α > 0⇔ [e −αd ij] ∈ E N ∀α > 0(895)where D = [d ij ] and E N is the elliptope (880).5.10.0.0.1 Proof. (Laurent, 2003) [233] (confer [165])Lemma 2.1. from A Tour d’Horizon ...on Completion Problems. [170]The following assertions are equivalent: for D=[d ij , i,j=1... N]∈ S N h andE N the elliptope in S N (5.9.1.0.1),(i) D ∈ EDM N(ii) e −αD = ∆ [e −αd ij] ∈ E N for all α > 0(iii) 11 T − e −αD ∆ = [1 − e −αd ij] ∈ EDM N for all α > 0⋄1) Equivalence of Lemma 2.1 (i) (ii) is stated in Schoenberg’s Theorem 1[233, p.527].2) (ii) ⇒ (iii) can be seen from the statement in the beginning of section 3,saying that a distance space embeds in L 2 iff some associated matrixis PSD. We reformulate it:Let d =(d ij ) i,j=0,1...N be a distance space on N+1 points(i.e., symmetric hollow matrix of order N+1) and let p =(p ij ) i,j=1...Nbe the symmetric matrix of order N related by:(A) 2p ij = d 0i + d 0j − d ij for i,j = 1... Nor equivalently(B) d 0i = p ii ,d ij = p ii + p jj − 2p ij for i,j = 1... NThen d embeds in L 2 iff p is a positive semidefinite matrix iff d is ofnegative type (second half page 525/top of page 526 in [233]).