v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
504 CHAPTER 6. CONE OF DISTANCE MATRICESby (913). Substituting this into EDM definition (1185), we get theHayden, Wells, Liu, & Tarazaga EDM formula [185,2]whereD(V X , y) y1 T + 1y T + λ N 11T − 2V X V T X ∈ EDM N (1191)λ 2‖V X ‖ 2 F = 1 T δ(V X V T X )2 and y δ(V X V T X ) − λ2N 1 = V δ(V XV T X )(1192)and y=0 if and only if 1 is an eigenvector of EDM D . Scalar λ becomesan eigenvalue when corresponding eigenvector 1 exists. 6.4Then the particular dyad sum from (1191)y1 T + 1y T + λ N 11T ∈ S N⊥c (1193)must belong to the orthogonal complement of the geometric center subspace(p.731), whereas V X V T X ∈ SN c ∩ S N + (1187) belongs to the positive semidefinitecone in the geometric center subspace.Proof. We validate eigenvector 1 and eigenvalue λ .(⇒) Suppose 1 is an eigenvector of EDM D . Then becauseit followsV T X 1 = 0 (1194)D1 = δ(V X V T X )1T 1 + 1δ(V X V T X )T 1 = N δ(V X V T X ) + ‖V X ‖ 2 F 1⇒ δ(V X V T X ) ∝ 1 (1195)For some κ∈ R +δ(V X V T X ) T 1 = N κ = tr(V T X V X ) = ‖V X ‖ 2 F ⇒ δ(V X V T X ) = 1 N ‖V X ‖ 2 F1 (1196)so y=0.(⇐) Now suppose δ(V X VX T)= λ 1 ; id est, y=0. Then2ND = λ N 11T − 2V X V T X ∈ EDM N (1197)1 is an eigenvector with corresponding eigenvalue λ . 6.4 e.g., when X = I in EDM definition (891).
6.4. EDM DEFINITION IN 11 T 505N(1 T )δ(V X V T X )1V XFigure 139: Example of V X selection to make an EDM corresponding tocardinality N = 3 and affine dimension r = 1 ; V X is a vector in nullspaceN(1 T )⊂ R 3 . Nullspace of 1 T is hyperplane in R 3 (drawn truncated) havingnormal 1. Vector δ(V X VX T) may or may not be in plane spanned by {1, V X } ,but belongs to nonnegative orthant which is strictly supported by N(1 T ).
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- Page 471 and 472: 5.11. EDM INDEFINITENESS 471we have
- Page 473 and 474: 5.11. EDM INDEFINITENESS 473So beca
- Page 475 and 476: 5.11. EDM INDEFINITENESS 475holds o
- Page 477 and 478: 5.12. LIST RECONSTRUCTION 477where
- Page 479 and 480: 5.12. LIST RECONSTRUCTION 479(a)(c)
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- Page 497 and 498: Chapter 6Cone of distance matricesF
- Page 499 and 500: 6.1. DEFINING EDM CONE 4996.1 Defin
- Page 501 and 502: 6.2. POLYHEDRAL BOUNDS 501This cone
- Page 503: 6.4. EDM DEFINITION IN 11 T 503That
- Page 507 and 508: 6.4. EDM DEFINITION IN 11 T 507Then
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- Page 529 and 530: 6.8. DUAL EDM CONE 529to the geomet
- Page 531 and 532: 6.8. DUAL EDM CONE 531Proof. First,
- Page 533 and 534: 6.8. DUAL EDM CONE 533EDM 2 = S 2 h
- Page 535 and 536: 6.8. DUAL EDM CONE 535therefore the
- Page 537 and 538: 6.8. DUAL EDM CONE 537Elegance of t
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- Page 543 and 544: 6.8. DUAL EDM CONE 5430dvec rel ∂
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- Page 549 and 550: 549project on the subspace, then pr
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6.4. EDM DEFINITION IN 11 T 505N(1 T )δ(V X V T X )1V XFigure 139: Example of V X selection to make an EDM corresponding tocardinality N = 3 and affine dimension r = 1 ; V X is a vector in nullspaceN(1 T )⊂ R 3 . Nullspace of 1 T is hyperplane in R 3 (drawn truncated) havingnormal 1. Vector δ(V X VX T) may or may not be in plane spanned by {1, V X } ,but belongs to nonnegative orthant which is strictly supported by N(1 T ).
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