v2006.03.09 - Convex Optimization

4.4. EDM DEFINITION 223The collection of all Euclidean distance matrices EDM N is a convex subsetof R N×N+ called the EDM cone (5, Figure 89, p.390);0 ∈ EDM N ⊆ S N h ∩ R N×N+ ⊂ S N (459)An EDM D must be expressible as a function of some list X ; id est, it musthave the formD(X) ∆ = δ(X T X)1 T + 1δ(X T X) T − 2X T X ∈ EDM N (460)= [vec(X) T (Φ ij ⊗ I) vecX , i,j=1... N] (461)Function D(X) will make an EDM given any X ∈ R n×N , conversely, butD(X) is not a convex function of X (4.4.1). Now the EDM cone may bedescribed:EDM N = { D(X) | X ∈ R N−1×N} (462)Expression D(X) is a matrix definition of EDM and so conforms to theEuclidean metric properties:Nonnegativity of EDM entries (property 1,4.2) is obvious from thedistance-square definition (456), so holds for any D expressible in the formD(X) in (460).When we say D is an EDM, reading from (460), it implicitly meansthe main diagonal must be 0 (property 2, self-distance) and D must besymmetric (property 3); δ(D) = 0 and D T = D or, equivalently, D ∈ S N hare necessary matrix criteria.4.4.0.1 homogeneityFunction D(X) is homogeneous in the sense, for ζ ∈ R√ √◦D(ζX) = |ζ|◦D(X) (463)where the positive square root is entrywise.Any nonnegatively scaled EDM remains an EDM; id est, the matrix classEDM is invariant to nonnegative scaling (αD(X) for α≥0) because allEDMs of dimension N constitute a convex cone EDM N (5, Figure 75).