v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
482 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXλ(−V T N OV N) j90080070060050040030020010001 2 3 4 5 6 7 8 9 10jFigure 135: Largest ten eigenvalues, of −V T N OV N for map of USA, sorted bynonincreasing value.5.13.2.1 Isotonic cartographyTo test Borg & Groenen’s conjecture, suppose we make a complete sort-indexmatrix O ∈ S N h ∩ R N×N+ for the map of the USA and then substitute O in placeof EDM D in the reconstruction process of5.12. Whereas EDM D returnedonly three significant eigenvalues (1140), the sort-index matrix O is generallynot an EDM (certainly not an EDM with corresponding affine dimension 3)so returns many more. The eigenvalues, calculated with absolute numericalerror approximately 5E-7, are plotted in Figure 135: (In the code onWıκımization, matrix O is normalized by (N(N −1)/2) 2 .)λ(−V T N OV N ) = [880.1 463.9 186.1 46.20 17.12 9.625 8.257 1.701 0.7128 0.6460 · · · ] T(1145)The extra eigenvalues indicate that affine dimension corresponding to anEDM near O is likely to exceed 3. To realize the map, we must simultaneouslyreduce that dimensionality and find an EDM D closest to O in somesense 5.58 while maintaining the known comparative distance relationship.For example: given permutation matrix Π expressing the known sortingaction like (1142) on entries5.58 a problem explored more in7.
5.13. RECONSTRUCTION EXAMPLES 483d 1 √2dvec D =⎡⎢⎣⎤d 12d 13d 23d 14d 24d 34⎥⎦.d N−1,N∈ R N(N−1)/2 (1146)of unknown D ∈ S N h , we can make sort-index matrix O input to theoptimization problemminimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3(1147)Πd ∈ K M+D ∈ EDM Nthat finds the EDM D (corresponding to affine dimension not exceeding 3 inisomorphic dvec EDM N ∩ Π T K M+ ) closest to O in the sense of Schoenberg(910).Analytical solution to this problem, ignoring the sort constraintΠd ∈ K M+ , is known [353]: we get the convex optimization [sic] (7.1)minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3(1148)D ∈ EDM NOnly the three largest nonnegative eigenvalues in (1145) need be retained tomake list (1132); the rest are discarded. The reconstruction from EDM Dfound in this manner is plotted in Figure 134e-f. Matlab code is onWıκımization. From these plots it becomes obvious that inclusion of thesort constraint is necessary for isotonic reconstruction.That sort constraint demands: any optimal solution D ⋆ must possess theknown comparative distance relationship that produces the original ordinaldistance data O (1144). Ignoring the sort constraint, apparently, violates it.Yet even more remarkable is how much the map reconstructed using onlyordinal data still resembles the original map of the USA after suffering themany violations produced by solving relaxed problem (1148). This suggeststhe simple reconstruction techniques of5.12 are robust to a significantamount of noise.
- 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 and 442: 5.6. INJECTIVITY OF D & UNIQUE RECO
- Page 443 and 444: 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: 5.13. RECONSTRUCTION EXAMPLES 481Wi
- 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
- Page 493 and 494: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 495 and 496: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- 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
- Page 509 and 510: 6.4. EDM DEFINITION IN 11 T 5096.4.
- Page 511 and 512: 6.5. CORRESPONDENCE TO PSD CONE S N
- Page 513 and 514: 6.5. CORRESPONDENCE TO PSD CONE S N
- Page 515 and 516: 6.5. CORRESPONDENCE TO PSD CONE S N
- Page 517 and 518: 6.6. VECTORIZATION & PROJECTION INT
- Page 519 and 520: 6.6. VECTORIZATION & PROJECTION INT
- Page 521 and 522: 6.6. VECTORIZATION & PROJECTION INT
- Page 523 and 524: 6.7. A GEOMETRY OF COMPLETION 523(b
- Page 525 and 526: 6.7. A GEOMETRY OF COMPLETION 525[3
- Page 527 and 528: 6.7. A GEOMETRY OF COMPLETION 527wh
- Page 529 and 530: 6.8. DUAL EDM CONE 529to the geomet
- Page 531 and 532: 6.8. DUAL EDM CONE 531Proof. First,
482 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXλ(−V T N OV N) j90080070060050040030020010001 2 3 4 5 6 7 8 9 10jFigure 135: Largest ten eigenvalues, of −V T N OV N for map of USA, sorted bynonincreasing value.5.13.2.1 Isotonic cartographyTo test Borg & Groenen’s conjecture, suppose we make a complete sort-indexmatrix O ∈ S N h ∩ R N×N+ for the map of the USA and then substitute O in placeof EDM D in the reconstruction process of5.12. Whereas EDM D returnedonly three significant eigenvalues (1140), the sort-index matrix O is generallynot an EDM (certainly not an EDM with corresponding affine dimension 3)so returns many more. The eigenvalues, calculated with absolute numericalerror approximately 5E-7, are plotted in Figure 135: (In the code onWıκımization, matrix O is normalized by (N(N −1)/2) 2 .)λ(−V T N OV N ) = [880.1 463.9 186.1 46.20 17.12 9.625 8.257 1.701 0.7128 0.6460 · · · ] T(1145)The extra eigenvalues indicate that affine dimension corresponding to anEDM near O is likely to exceed 3. To realize the map, we must simultaneouslyreduce that dimensionality and find an EDM D closest to O in somesense 5.58 while maintaining the known comparative distance relationship.For example: given permutation matrix Π expressing the known sortingaction like (1142) on entries5.58 a problem explored more in7.
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