v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
466 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXsmallest eigenvalue1 2 3 4d 14-0.2-0.4-0.6Figure 132: Smallest eigenvalue of −VN TDV Nonly one value of d 14 : 2.makes it a PSD matrix for1.41.2d 1/αijlog 2 (1+d 1/αij )(a)10.80.60.40.21−e −αd ij40.5 1 1.5 2d ij3(b)21d 1/αijlog 2 (1+d 1/αij )1−e −αd ij0.5 1 1.5 2αFigure 133: Some entrywise EDM compositions: (a) α = 2. Concavenondecreasing in d ij . (b) Trajectory convergence in α for d ij = 2.
5.10. EDM-ENTRY COMPOSITION 467Then d embeds in L 2 iff p is a positive semidefinite matrix iff dis of negative type (second half page 525/top of page 526 in [313]).For the implication from (ii) to (iii), set: p = e −αd and define d ′from p using (B) above. Then d ′ is a distance space on N+1points that embeds in L 2 . Thus its subspace of N points alsoembeds in L 2 and is precisely 1 − e −αd .Note that (iii) ⇒ (ii) cannot be read immediately from this argumentsince (iii) involves the subdistance of d ′ on N points (and not the fulld ′ on N+1 points).3) Show (iii) ⇒ (i) by using the series expansion of the function 1 − e −αd :the constant term cancels, α factors out; there remains a summationof d plus a multiple of α . Letting α go to 0 gives the result.This is not explicitly written in Schoenberg, but he also uses suchan argument; expansion of the exponential function then α → 0 (firstproof on [313, p.526]).Schoenberg’s results [313,6 thm.5] (confer [227, p.108-109]) also suggestcertain finite positive roots of EDM entries produce EDMs; specifically,D ∈ EDM N ⇔ [d 1/αij ] ∈ EDM N ∀α > 1 (1093)The special case α = 2 is of interest because it corresponds to absolutedistance; e.g.,D ∈ EDM N ⇒ ◦√ D ∈ EDM N (1094)Assuming that points constituting a corresponding list X are distinct(1055), then it follows: for D ∈ S N hlimα→∞ [d1/α ij] = limα→∞[1 − e −αd ij] = −E 11 T − I (1095)Negative elementary matrix −E (B.3) is relatively interior to the EDM cone(6.5) and terminal to respective trajectories (1092a) and (1093) as functionsof α . Both trajectories are confined to the EDM cone; in engineering terms,the EDM cone is an invariant set [310] to either trajectory. Further, if D isnot an EDM but for some particular α p it becomes an EDM, then for allgreater values of α it remains an EDM.
- 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
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- Page 445 and 446: 5.7. EMBEDDING IN AFFINE HULL 4455.
- Page 447 and 448: 5.7. EMBEDDING IN AFFINE HULL 447Fo
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- Page 465: 5.10. EDM-ENTRY COMPOSITION 465of a
- 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
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- Page 485 and 486: 5.13. RECONSTRUCTION EXAMPLES 485Th
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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 and 504: 6.4. EDM DEFINITION IN 11 T 503That
- Page 505 and 506: 6.4. EDM DEFINITION IN 11 T 505N(1
- Page 507 and 508: 6.4. EDM DEFINITION IN 11 T 507Then
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466 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXsmallest eigenvalue1 2 3 4d 14-0.2-0.4-0.6Figure 132: Smallest eigenvalue of −VN TDV Nonly one value of d 14 : 2.makes it a PSD matrix for1.41.2d 1/αijlog 2 (1+d 1/αij )(a)10.80.60.40.21−e −αd ij40.5 1 1.5 2d ij3(b)21d 1/αijlog 2 (1+d 1/αij )1−e −αd ij0.5 1 1.5 2αFigure 133: Some entrywise EDM compositions: (a) α = 2. Concavenondecreasing in d ij . (b) Trajectory convergence in α for d ij = 2.
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