v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 353
The collection of all Euclidean distance matrices EDM N is a convex subset
of R N×N
+ called the EDM cone (6, Figure 128, p.498);
0 ∈ EDM N ⊆ S N h ∩ R N×N
+ ⊂ S N (797)
An EDM D must be expressible as a function of some list X ; id est, it must
have the form
D(X) ∆ = δ(X T X)1 T + 1δ(X T X) T − 2X T X ∈ EDM N (798)
= [vec(X) T (Φ ij ⊗ I) vecX , i,j=1... N] (799)
Function D(X) will make an EDM given any X ∈ R n×N , conversely, but
D(X) is not a convex function of X (5.4.1). Now the EDM cone may be
described:
EDM N = { D(X) | X ∈ R N−1×N} (800)
Expression D(X) is a matrix definition of EDM and so conforms to the
Euclidean metric properties:
Nonnegativity of EDM entries (property 1,5.2) is obvious from the
distance-square definition (794), so holds for any D expressible in the form
D(X) in (798).
When we say D is an EDM, reading from (798), it implicitly means
the main diagonal must be 0 (property 2, self-distance) and D must be
symmetric (property 3); δ(D) = 0 and D T = D or, equivalently, D ∈ S N h
are necessary matrix criteria.
5.4.0.1 homogeneity
Function D(X) is homogeneous in the sense, for ζ ∈ R
√ √
◦
D(ζX) = |ζ|
◦
D(X) (801)
where the positive square root is entrywise.
Any nonnegatively scaled EDM remains an EDM; id est, the matrix class
EDM is invariant to nonnegative scaling (αD(X) for α≥0) because all
EDMs of dimension N constitute a convex cone EDM N (6, Figure 115).