v2009.01.01 - Convex Optimization

424 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
5.12 List reconstruction
The traditional term metric multidimensional scaling 5.51 [221] [90] [306]
[88] [224] [78] refers to any reconstruction of a list X ∈ R n×N in
Euclidean space from interpoint distance information, possibly incomplete
(6.4), ordinal (5.13.2), or specified perhaps only by bounding-constraints
(5.4.2.2.8) [304]. Techniques for reconstruction are essentially methods
for optimally embedding an unknown list of points, corresponding to
given Euclidean distance data, in an affine subset of desired or minimum
dimension. The oldest known precursor is called principal component
analysis [139] which analyzes the correlation matrix (5.9.1.0.1); [46,22]
a.k.a, Karhunen−Loéve transform in the digital signal processing literature.
Isometric reconstruction (5.5.3) of point list X is best performed by eigen
decomposition of a Gram matrix; for then, numerical errors of factorization
are easily spotted in the eigenvalues: Now we consider how rotation/reflection
and translation invariance factor into a reconstruction.
5.12.1 x 1 at the origin. V N
At the stage of reconstruction, we have D ∈ EDM N and we wish to find
a generating list (2.3.2) for P − α by factoring positive semidefinite
−VN TDV N (995) as suggested in5.9.1.0.5. One way to factor −VN TDV N
is via diagonalization of symmetric matrices; [287,5.6] [176] (A.5.2,A.3)
−VNDV T ∆ N = QΛQ T (1033)
QΛQ T ≽ 0 ⇔ Λ ≽ 0 (1034)
where Q∈ R N−1×N−1 is an orthogonal matrix containing eigenvectors
while Λ∈ S N−1 is a diagonal matrix containing corresponding nonnegative
eigenvalues ordered by nonincreasing value. From the diagonalization,
identify the list using (940);
−V T NDV N = 2V T NX T XV N ∆ = Q √ ΛQ T pQ p
√
ΛQ
T
(1035)
5.51 Scaling [302] means making a scale, i.e., a numerical representation of qualitative data.
If the scale is multidimensional, it’s multidimensional scaling.
−Jan de Leeuw
When data comprises measurable distances, then reconstruction is termed metric
multidimensional scaling. In one dimension, N coordinates in X define the scale.