v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

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476 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.12 List reconstructionThe term metric multidimensional scaling 5.55 [257] [106] [353] [104] [260][89] refers to any reconstruction of a list X ∈ R n×N in Euclidean spacefrom interpoint distance information, possibly incomplete (6.7), ordinal(5.13.2), or specified perhaps only by bounding-constraints (5.4.2.3.9)[351]. Techniques for reconstruction are essentially methods for optimallyembedding an unknown list of points, corresponding to given Euclideandistance data, in an affine subset of desired or minimum dimension.The oldest known precursor is called principal component analysis [165]which analyzes the correlation matrix (5.9.1.0.1); [53,22] a.k.a,Karhunen-Loéve transform in the digital signal processing literature.Isometric reconstruction (5.5.3) of point list X is best performed byeigenvalue decomposition of a Gram matrix; for then, numerical errors offactorization are easily spotted in the eigenvalues: Now we consider howrotation/reflection and translation invariance factor into a reconstruction.5.12.1 x 1 at the origin. V NAt the stage of reconstruction, we have D ∈ EDM N and wish to find agenerating list (2.3.2) for polyhedron P − α by factoring Gram matrix−V T N DV N (908) as in (1090). One way to factor −V T N DV N is viadiagonalization of symmetric matrices; [331,5.6] [202] (A.5.1,A.3)−V T NDV N QΛQ T (1128)QΛQ T ≽ 0 ⇔ Λ ≽ 0 (1129)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 (1035);−V T NDV N = 2V T NX T XV N Q √ ΛQ T pQ p√ΛQT(1130)5.55 Scaling [349] means making a scale, i.e., a numerical representation of qualitative data.If the scale is multidimensional, it’s multidimensional scaling.−Jan de LeeuwA goal of multidimensional scaling is to find a low-dimensional representation of listX so that distances between its elements best fit a given set of measured pairwisedissimilarities. When data comprises measurable distances, then reconstruction is termedmetric multidimensional scaling. In one dimension, N coordinates in X define the scale.

5.12. LIST RECONSTRUCTION 477where √ ΛQ T pQ p√Λ Λ =√Λ√Λ and where Qp ∈ R n×N−1 is unknown as isits dimension n . Rotation/reflection is accounted for by Q p yet only itsfirst r columns are necessarily orthonormal. 5.56 Assuming membership tothe unit simplex y ∈ S (1087), then point p = X √ 2V N y = Q p√ΛQ T y in R nbelongs to the translated polyhedronP − x 1 (1131)whose generating list constitutes the columns of (1029)[ √ ] [ √ ]0 X 2VN = 0 Q p ΛQT∈ R n×N= [0 x 2 −x 1 x 3 −x 1 · · · x N −x 1 ](1132)The scaled auxiliary matrix V N represents that translation. A simple choicefor Q p has n set to N − 1; id est, Q p = I . Ideally, each member of thegenerating list has at most r nonzero entries; r being, affine dimensionrankV T NDV N = rankQ p√ΛQ T = rank Λ = r (1133)Each member then has at least N −1 − r zeros in its higher-dimensionalcoordinates because r ≤ N −1. (1041) To truncate those zeros, choose nequal to affine dimension which is the smallest n possible because XV N hasrank r ≤ n (1037). 5.57 In that case, the simplest choice for Q p is [I 0 ]having dimension r ×N −1.We may wish to verify the list (1132) found from the diagonalization of−VN TDV N . Because of rotation/reflection and translation invariance (5.5),EDM D can be uniquely made from that list by calculating: (891)5.56 Recall r signifies affine dimension. Q p is not necessarily an orthogonal matrix. Q p isconstrained such that only its first r columns are necessarily orthonormal because thereare only r nonzero eigenvalues in Λ when −VN TDV N has rank r (5.7.1.1). Remainingcolumns of Q p are arbitrary.⎡⎤⎡q T1 ⎤λ 1 0. .. 5.57 If we write Q T = ⎣. .. ⎦ as rowwise eigenvectors, Λ = ⎢ λr ⎥qN−1T ⎣ 0 ... ⎦0 0in terms of eigenvalues, and Q p = [ ]q p1 · · · q pN−1 as column vectors, then√Q p Λ Q T ∑= r √λi q pi qiT is a sum of r linearly independent rank-one matrices (B.1.1).i=1Hence the summation has rank r .

476 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.12 List reconstructionThe term metric multidimensional scaling 5.55 [257] [106] [353] [104] [260][89] refers to any reconstruction of a list X ∈ R n×N in Euclidean spacefrom interpoint distance information, possibly incomplete (6.7), ordinal(5.13.2), or specified perhaps only by bounding-constraints (5.4.2.3.9)[351]. Techniques for reconstruction are essentially methods for optimallyembedding an unknown list of points, corresponding to given Euclideandistance data, in an affine subset of desired or minimum dimension.The oldest known precursor is called principal component analysis [165]which analyzes the correlation matrix (5.9.1.0.1); [53,22] a.k.a,Karhunen-Loéve transform in the digital signal processing literature.Isometric reconstruction (5.5.3) of point list X is best performed byeigenvalue decomposition of a Gram matrix; for then, numerical errors offactorization are easily spotted in the eigenvalues: Now we consider howrotation/reflection and translation invariance factor into a reconstruction.5.12.1 x 1 at the origin. V NAt the stage of reconstruction, we have D ∈ EDM N and wish to find agenerating list (2.3.2) for polyhedron P − α by factoring Gram matrix−V T N DV N (908) as in (1090). One way to factor −V T N DV N is viadiagonalization of symmetric matrices; [331,5.6] [202] (A.5.1,A.3)−V T NDV N QΛQ T (1128)QΛQ T ≽ 0 ⇔ Λ ≽ 0 (1129)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 (1035);−V T NDV N = 2V T NX T XV N Q √ ΛQ T pQ p√ΛQT(1130)5.55 Scaling [349] means making a scale, i.e., a numerical representation of qualitative data.If the scale is multidimensional, it’s multidimensional scaling.−Jan de LeeuwA goal of multidimensional scaling is to find a low-dimensional representation of listX so that distances between its elements best fit a given set of measured pairwisedissimilarities. When data comprises measurable distances, then reconstruction is termedmetric multidimensional scaling. In one dimension, N coordinates in X define the scale.

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