v2007.09.13 - Convex Optimization

366 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.12 List reconstructionThe traditional term multidimensional scaling 5.50 [187] [71] [262] [69][190] [61] refers to any reconstruction of a list X ∈ R n×N in Euclideanspace from interpoint distance information, possibly incomplete (6.4),ordinal (5.13.2), or specified perhaps only by bounding-constraints(5.4.2.2.7) [263]. Techniques for reconstruction are essentially methodsfor optimally embedding an unknown list of points, corresponding togiven Euclidean distance data, in an affine subset of desired or minimumdimension. The oldest known precursor is called principal componentanalysis [114] which analyzes the correlation matrix (5.9.1.0.1); [39,22]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 eigendecomposition of a Gram matrix; for then, numerical errors of factorizationare easily spotted in the eigenvalues.We now consider how rotation/reflection and translation invariance factorinto a reconstruction.5.12.1 x 1 at the origin. V NAt the stage of reconstruction, we have D ∈ EDM N and we wish to finda generating list (2.3.2) for P − α by factoring positive semidefinite−VN TDV N (893) as suggested in5.9.1.0.4. One way to factor −VN TDV Nis via diagonalization of symmetric matrices; [247,5.6] [149] (A.5.2,A.3)−VNDV T ∆ N = QΛQ T (931)QΛQ T ≽ 0 ⇔ Λ ≽ 0 (932)where Q∈ R N−1×N−1 is an orthogonal matrix containing eigenvectorswhile Λ∈ S N−1 is a diagonal matrix containing corresponding nonnegativeeigenvalues ordered by nonincreasing value. From the diagonalization,identify the list using (840);−V T NDV N = 2V T NX T XV N ∆ = Q √ ΛQ T pQ p√ΛQT(933)5.50 Scaling [260] means making a scale, i.e., a numerical representation of qualitative data.If the scale is multidimensional, it’s multidimensional scaling.−Jan de LeeuwWhen the metric is Euclidean distance, then reconstruction is termed metricmultidimensional scaling.