v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
368 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXD(X) = D(X[0 √ 2V N ]) = D(Q p [0 √ ΛQ T ]) = D([0 √ ΛQ T ]) (937)This suggests a way to find EDM D given −V T N DV N ; (confer (818))[0D =δ ( −VN TDV )N] [1 T + 1 0 δ ( −VN TDV ) ] TN[ 0 0T− 20 −VN TDV N(725)]5.12.2 0 geometric center. VAlternatively, we may perform reconstruction using instead the auxiliarymatrix V (B.4.1), corresponding to the polyhedronP − α c (938)whose geometric center has been translated to the origin. Redimensioningdiagonalization factors Q, Λ∈R N×N and unknown Q p ∈ R n×N , (841)−V DV = 2V X T XV ∆ = Q √ ΛQ T pQ p√ΛQT ∆ = QΛQ T (939)where the geometrically centered generating list constitutes (confer (935))XV = 1 √2Q p√ΛQ T ∈ R n×N= [x 1 − 1 N X1 x 2 − 1 N X1 x 3 − 1 N X1 · · · x N − 1 N X1 ] (940)where α c = 1 N X1. (5.5.1.0.1) The simplest choice for Q p is [I 0 ]∈ R r×N .Now EDM D can be uniquely made from the list found, by calculating:(705)D(X) = D(XV ) = D( 1 √2Q p√ΛQ T ) = D( √ ΛQ T ) 1 2(941)This EDM is, of course, identical to (937). Similarly to (725), from −V DVwe can find EDM D ; (confer (805))D = δ(−V DV 1 2 )1T + 1δ(−V DV 1 2 )T − 2(−V DV 1 2 ) (731)
5.12. LIST RECONSTRUCTION 369(a)(c)(b)(d)(f)(e)Figure 90: Map of United States of America showing some state boundariesand the Great Lakes. All plots made using 5020 connected points. Anydifference in scale in (a) through (d) is an artifact of plotting routine.(a) shows original map made from decimated (latitude, longitude) data.(b) Original map data rotated (freehand) to highlight curvature of Earth.(c) Map isometrically reconstructed from the EDM.(d) Same reconstructed map illustrating curvature.(e)(f) Two views of one isotonic reconstruction; problem (950) with no sortconstraint Πd (and no hidden line removal).
- Page 317 and 318: 5.4. EDM DEFINITION 317corrected by
- Page 319 and 320: 5.4. EDM DEFINITION 319aptly be app
- Page 321 and 322: 5.4. EDM DEFINITION 321As before, a
- Page 323 and 324: 5.4. EDM DEFINITION 323where ([√t
- Page 325 and 326: 5.4. EDM DEFINITION 325because (A.3
- Page 327 and 328: 5.5. INVARIANCE 3275.5.1.0.1 Exampl
- Page 329 and 330: 5.6. INJECTIVITY OF D & UNIQUE RECO
- Page 331 and 332: 5.6. INJECTIVITY OF D & UNIQUE RECO
- Page 333 and 334: 5.6. INJECTIVITY OF D & UNIQUE RECO
- Page 335 and 336: 5.7. EMBEDDING IN AFFINE HULL 3355.
- Page 337 and 338: 5.7. EMBEDDING IN AFFINE HULL 337Fo
- Page 339 and 340: 5.7. EMBEDDING IN AFFINE HULL 3395.
- Page 341 and 342: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
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- Page 349 and 350: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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- Page 357 and 358: 5.10. EDM-ENTRY COMPOSITION 357(ii)
- Page 359 and 360: 5.11. EDM INDEFINITENESS 3595.11.1
- Page 361 and 362: 5.11. EDM INDEFINITENESS 361(confer
- Page 363 and 364: 5.11. EDM INDEFINITENESS 363we have
- Page 365 and 366: 5.11. EDM INDEFINITENESS 365For pre
- Page 367: 5.12. LIST RECONSTRUCTION 367where
- Page 371 and 372: 5.13. RECONSTRUCTION EXAMPLES 371D
- Page 373 and 374: 5.13. RECONSTRUCTION EXAMPLES 373Th
- Page 375 and 376: 5.13. RECONSTRUCTION EXAMPLES 375wh
- Page 377 and 378: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 379 and 380: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
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- Page 385 and 386: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 387 and 388: Chapter 6EDM coneFor N > 3, the con
- Page 389 and 390: 6.1. DEFINING EDM CONE 3896.1 Defin
- Page 391 and 392: 6.2. POLYHEDRAL BOUNDS 391This cone
- Page 393 and 394: 6.3.√EDM CONE IS NOT CONVEX 393N
- Page 395 and 396: 6.4. A GEOMETRY OF COMPLETION 3956.
- Page 397 and 398: 6.4. A GEOMETRY OF COMPLETION 397(a
- Page 399 and 400: 6.4. A GEOMETRY OF COMPLETION 399Fi
- Page 401 and 402: 6.5. EDM DEFINITION IN 11 T 401and
- Page 403 and 404: 6.5. EDM DEFINITION IN 11 T 403then
- Page 405 and 406: 6.5. EDM DEFINITION IN 11 T 4056.5.
- Page 407 and 408: 6.5. EDM DEFINITION IN 11 T 407D =
- Page 409 and 410: 6.6. CORRESPONDENCE TO PSD CONE S N
- Page 411 and 412: 6.6. CORRESPONDENCE TO PSD CONE S N
- Page 413 and 414: 6.6. CORRESPONDENCE TO PSD CONE S N
- Page 415 and 416: 6.7. VECTORIZATION & PROJECTION INT
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5.12. LIST RECONSTRUCTION 369(a)(c)(b)(d)(f)(e)Figure 90: Map of United States of America showing some state boundariesand the Great Lakes. All plots made using 5020 connected points. Anydifference in scale in (a) through (d) is an artifact of plotting routine.(a) shows original map made from decimated (latitude, longitude) data.(b) Original map data rotated (freehand) to highlight curvature of Earth.(c) Map isometrically reconstructed from the EDM.(d) Same reconstructed map illustrating curvature.(e)(f) Two views of one isotonic reconstruction; problem (950) with no sortconstraint Πd (and no hidden line removal).
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