v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
296 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXNow we develop some invaluable concepts, moving toward a link of theEuclidean metric properties to matrix criteria.5.4 EDM definitionAscribe points in a list {x l ∈ R n , l=1... N} to the columns of a matrixX = [x 1 · · · x N ] ∈ R n×N (65)where N is regarded as cardinality of list X . When matrix D =[d ij ] is anEDM, its entries must be related to those points constituting the list by theEuclidean distance-square: for i,j=1... N (A.1.1 no.23)d ij = ‖x i − x j ‖ 2 = (x i − x j ) T (x i − x j ) = ‖x i ‖ 2 + ‖x j ‖ 2 − 2x T ix j= [ ] [ ] [ ]x T i x T I −I xij−I I x j= vec(X) T (Φ ij ⊗ I) vec X = 〈Φ ij , X T X〉(701)where⎡vec X = ⎢⎣⎤x 1x 2⎥. ⎦ ∈ RnN (702)x Nand where Φ ij ⊗ I has I ∈ S n in its ii th and jj th block of entries while−I ∈ S n fills its ij th and ji th block; id est,Φ ij ∆ = δ((e i e T j + e j e T i )1) − (e i e T j + e j e T i ) ∈ S N += e i e T i + e j e T j − e i e T j − e j e T (703)i= (e i − e j )(e i − e j ) Twhere {e i ∈ R N , i=1... N} is the set of standard basis vectors, and ⊗signifies the Kronecker product (D.1.2.1). Thus each entry d ij is a convexquadratic function [46,3,4] of vec X (30). [228,6]
5.4. EDM DEFINITION 297The collection of all Euclidean distance matrices EDM N is a convex subsetof R N×N+ called the EDM cone (6, Figure 108, p.440);0 ∈ EDM N ⊆ S N h ∩ R N×N+ ⊂ S N (704)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 (705)= [vec(X) T (Φ ij ⊗ I) vecX , i,j=1... N] (706)Function D(X) will make an EDM given any X ∈ R n×N , conversely, butD(X) is not a convex function of X (5.4.1). Now the EDM cone may bedescribed:EDM N = { D(X) | X ∈ R N−1×N} (707)Expression D(X) is a matrix definition of EDM and so conforms to theEuclidean metric properties:Nonnegativity of EDM entries (property 1,5.2) is obvious from thedistance-square definition (701), so holds for any D expressible in the formD(X) in (705).When we say D is an EDM, reading from (705), 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.5.4.0.1 homogeneityFunction D(X) is homogeneous in the sense, for ζ ∈ R√ √◦D(ζX) = |ζ|◦D(X) (708)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 (6, Figure 95).
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5.4. EDM DEFINITION 297The collection of all Euclidean distance matrices EDM N is a convex subsetof R N×N+ called the EDM cone (6, Figure 108, p.440);0 ∈ EDM N ⊆ S N h ∩ R N×N+ ⊂ S N (704)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 (705)= [vec(X) T (Φ ij ⊗ I) vecX , i,j=1... N] (706)Function D(X) will make an EDM given any X ∈ R n×N , conversely, butD(X) is not a convex function of X (5.4.1). Now the EDM cone may bedescribed:EDM N = { D(X) | X ∈ R N−1×N} (707)Expression D(X) is a matrix definition of EDM and so conforms to theEuclidean metric properties:Nonnegativity of EDM entries (property 1,5.2) is obvious from thedistance-square definition (701), so holds for any D expressible in the formD(X) in (705).When we say D is an EDM, reading from (705), 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.5.4.0.1 homogeneityFunction D(X) is homogeneous in the sense, for ζ ∈ R√ √◦D(ζX) = |ζ|◦D(X) (708)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 (6, Figure 95).