v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
442 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXbecause the projection of −D/2 on S N c (1998) can be 0 if and only ifD ∈ S N⊥c ; but S N⊥c ∩ S N h = 0 (Figure 129). Projector V on S N h is thereforeinjective hence uniquely invertible. Further, −V S N h V/2 is equivalent to thegeometric center subspace S N c in the ambient space of symmetric matrices; asurjection,S N c = V(S N ) = V ( S N h ⊕ S N⊥h) ( )= V SNh(997)because (72)V ( ) ( ) (S N h ⊇ V SN⊥h = V δ 2 (S N ) ) (998)Because D(G) on S N c is injective, and aff D ( V(EDM N ) ) = D ( V(aff EDM N ) )by property (127) of the affine hull, we find for D ∈ S N hid est,orD(−V DV 1 2 ) = δ(−V DV 1 2 )1T + 1δ(−V DV 1 2 )T − 2(−V DV 1 2 ) (999)D = D ( V(D) ) (1000)−V DV = V ( D(−V DV ) ) (1001)S N h = D ( V(S N h ) ) (1002)−V S N h V = V ( D(−V S N h V ) ) (1003)These operators V and D are mutual inverses.The Gram-form D ( )S N c (903) is equivalent to SNh ;D ( S N c)= D(V(SNh ⊕ S N⊥h) ) = S N h + D ( V(S N⊥h ) ) = S N h (1004)because S N h ⊇ D ( V(S N⊥h ) ) . In summary, for the Gram-form we have theisomorphisms [90,2] [89, p.76, p.107] [7,2.1] 5.30 [6,2] [8,18.2.1] [2,2.1]and from bijectivity results in5.6.1,S N h = D(S N c ) (1005)S N c = V(S N h ) (1006)EDM N = D(S N c ∩ S N +) (1007)S N c ∩ S N + = V(EDM N ) (1008)5.30 In [7, p.6, line 20], delete sentence: Since G is also...not a singleton set.[7, p.10, line 11] x 3 =2 (not 1).
5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 4435.6.2 Inner-product form bijectivityThe Gram-form EDM operator D(G)= δ(G)1 T + 1δ(G) T − 2G (903) is aninjective map, for example, on the domain that is the subspace of symmetricmatrices having all zeros in the first row and columnS N 1 = {G∈ S N | Ge 1 = 0}{[ ] [ 0 0T 0 0T= Y0 I 0 I]| Y ∈ S N }(2002)because it obviously has no nullspace there. Since Ge 1 = 0 ⇔ Xe 1 = 0 (905)means the first point in the list X resides at the origin, then D(G) on S N 1 ∩ S N +must be surjective onto EDM N .Substituting Θ T Θ ← −VN TDV N (970) into inner-product form EDMdefinition D(Θ) (958), it may be further decomposed:[0D(D) =δ ( −VN TDV )N] [1 T + 1 0 δ ( −VN TDV ) ] TN[ ] 0 0T− 20 −VN TDV N(1009)This linear operator D is another flavor of inner-product form and an injectivemap of the EDM cone onto itself. Yet when its domain is instead the entiresymmetric hollow subspace S N h = aff EDM N , D(D) becomes an injectivemap onto that same subspace. Proof follows directly from the fact: linear Dhas no nullspace [85,A.1] on S N h = aff D(EDM N )= D(aff EDM N ) (127).5.6.2.1 Inversion of D ( −VN TDV )NInjectivity of D(D) suggests inversion of (confer (908))V N (D) : S N → S N−1 −V T NDV N (1010)a linear surjective 5.31 mapping onto S N−1 having nullspace 5.32 S N⊥c ;V N (S N h ) = S N−1 (1011)5.31 Surjectivity of V N (D) is demonstrated via the Gram-form EDM operator D(G):Since S N h = D(S N c ) (1004), then for any Y ∈ S N−1 , −VN T †TD(VN Y V † N /2)V N = Y .5.32 N(V N ) ⊇ S N⊥c is apparent. There exists a linear mappingT(V N (D)) V †TN V N(D)V † N = −V DV 1 2 = V(D)
- Page 391 and 392: 4.8. CONVEX ITERATION RANK-1 391whi
- Page 393 and 394: 4.8. CONVEX ITERATION RANK-1 393the
- Page 395 and 396: Chapter 5Euclidean Distance MatrixT
- Page 397 and 398: 5.2. FIRST METRIC PROPERTIES 397to
- Page 399 and 400: 5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
- Page 401 and 402: 5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
- Page 403 and 404: 5.4. EDM DEFINITION 403The collecti
- Page 405 and 406: 5.4. EDM DEFINITION 4055.4.2 Gram-f
- Page 407 and 408: 5.4. EDM DEFINITION 407We provide a
- Page 409 and 410: 5.4. EDM DEFINITION 4095.4.2.3.1 Ex
- Page 411 and 412: 5.4. EDM DEFINITION 411is the fact:
- Page 413 and 414: 5.4. EDM DEFINITION 413Figure 120:
- Page 415 and 416: 5.4. EDM DEFINITION 415is found fro
- Page 417 and 418: 5.4. EDM DEFINITION 417one less dim
- Page 419 and 420: 5.4. EDM DEFINITION 419equality con
- Page 421 and 422: 5.4. EDM DEFINITION 421How much dis
- Page 423 and 424: 5.4. EDM DEFINITION 423105ˇx 4ˇx
- Page 425 and 426: 5.4. EDM DEFINITION 425now implicit
- Page 427 and 428: 5.4. EDM DEFINITION 427by translate
- Page 429 and 430: 5.4. EDM DEFINITION 429Crippen & Ha
- Page 431 and 432: 5.4. EDM DEFINITION 431where ([√t
- Page 433 and 434: 5.4. EDM DEFINITION 433because (A.3
- Page 435 and 436: 5.5. INVARIANCE 4355.5.1.0.1 Exampl
- Page 437 and 438: 5.5. INVARIANCE 437x 2 x 2x 3 x 1 x
- Page 439 and 440: 5.6. INJECTIVITY OF D & UNIQUE RECO
- Page 441: 5.6. INJECTIVITY OF D & UNIQUE RECO
- Page 445 and 446: 5.7. EMBEDDING IN AFFINE HULL 4455.
- Page 447 and 448: 5.7. EMBEDDING IN AFFINE HULL 447Fo
- Page 449 and 450: 5.7. EMBEDDING IN AFFINE HULL 4495.
- Page 451 and 452: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 453 and 454: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 455 and 456: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 457 and 458: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 459 and 460: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 461 and 462: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 463 and 464: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 465 and 466: 5.10. EDM-ENTRY COMPOSITION 465of a
- Page 467 and 468: 5.10. EDM-ENTRY COMPOSITION 467Then
- Page 469 and 470: 5.11. EDM INDEFINITENESS 4695.11.1
- 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)
- Page 481 and 482: 5.13. RECONSTRUCTION EXAMPLES 481Wi
- Page 483 and 484: 5.13. RECONSTRUCTION EXAMPLES 483d
- Page 485 and 486: 5.13. RECONSTRUCTION EXAMPLES 485Th
- Page 487 and 488: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 489 and 490: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 491 and 492: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 4435.6.2 Inner-product form bijectivityThe Gram-form EDM operator D(G)= δ(G)1 T + 1δ(G) T − 2G (903) is aninjective map, for example, on the domain that is the subspace of symmetricmatrices having all zeros in the first row and columnS N 1 = {G∈ S N | Ge 1 = 0}{[ ] [ 0 0T 0 0T= Y0 I 0 I]| Y ∈ S N }(2002)because it obviously has no nullspace there. Since Ge 1 = 0 ⇔ Xe 1 = 0 (905)means the first point in the list X resides at the origin, then D(G) on S N 1 ∩ S N +must be surjective onto EDM N .Substituting Θ T Θ ← −VN TDV N (970) into inner-product form EDMdefinition D(Θ) (958), it may be further decomposed:[0D(D) =δ ( −VN TDV )N] [1 T + 1 0 δ ( −VN TDV ) ] TN[ ] 0 0T− 20 −VN TDV N(1009)This linear operator D is another flavor of inner-product form and an injectivemap of the EDM cone onto itself. Yet when its domain is instead the entiresymmetric hollow subspace S N h = aff EDM N , D(D) becomes an injectivemap onto that same subspace. Proof follows directly from the fact: linear Dhas no nullspace [85,A.1] on S N h = aff D(EDM N )= D(aff EDM N ) (127).5.6.2.1 Inversion of D ( −VN TDV )NInjectivity of D(D) suggests inversion of (confer (908))V N (D) : S N → S N−1 −V T NDV N (1010)a linear surjective 5.31 mapping onto S N−1 having nullspace 5.32 S N⊥c ;V N (S N h ) = S N−1 (1011)5.31 Surjectivity of V N (D) is demonstrated via the Gram-form EDM operator D(G):Since S N h = D(S N c ) (1004), then for any Y ∈ S N−1 , −VN T †TD(VN Y V † N /2)V N = Y .5.32 N(V N ) ⊇ S N⊥c is apparent. There exists a linear mappingT(V N (D)) V †TN V N(D)V † N = −V DV 1 2 = V(D)