v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
372 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXλ(−V T N OV N) j90080070060050040030020010001 2 3 4 5 6 7 8 9 10jFigure 91: Largest ten eigenvalues of −VN TOV N for map of USA, sorted bynonincreasing value. In the code (F.3.2), we normalize O by (N(N −1)/2) 2 .Any process of reconstruction that leaves comparative distanceinformation intact is called ordinal multidimensional scaling or isotonicreconstruction. Beyond rotation, reflection, and translation error, (5.5)list reconstruction by isotonic reconstruction is subject to error in absolutescale (dilation) and distance ratio. Yet Borg & Groenen argue: [39,2.2]reconstruction from complete comparative distance information for a largenumber of points is as highly constrained as reconstruction from an EDM;the larger the number, the better.5.13.2.1 Isotonic map of the USATo 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 (943), 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 91: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(948)
5.13. RECONSTRUCTION EXAMPLES 373The 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 some sense(a problem explored more in7) while maintaining the known comparativedistance relationship; e.g., given permutation matrix Π expressing theknown sorting action on the entries d of unknown D ∈ S N h , (63)d ∆ = 1 √2dvec D =⎡⎢⎣⎤d 12d 13d 23d 14d 24d 34⎥⎦.d N−1,N∈ R N(N−1)/2 (949)we can make the sort-index matrix O input to the optimization problemminimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3(950)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(724).Analytical solution to this problem, ignoring the sort constraintΠd ∈ K M+ , is known [262]: we get the convex optimization [sic] (7.1)minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3(951)D ∈ EDM NOnly the three largest nonnegative eigenvalues in (948) need be retainedto make list (935); the rest are discarded. The reconstruction fromEDM D found in this manner is plotted in Figure 90(e)(f) from which itbecomes obvious that inclusion of the sort constraint is necessary for isotonicreconstruction.
- 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
- Page 343 and 344: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 345 and 346: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 347 and 348: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 349 and 350: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 351 and 352: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 353 and 354: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- Page 355 and 356: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
- 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 and 368: 5.12. LIST RECONSTRUCTION 367where
- Page 369 and 370: 5.12. LIST RECONSTRUCTION 369(a)(c)
- Page 371: 5.13. RECONSTRUCTION EXAMPLES 371D
- 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
- Page 381 and 382: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- Page 383 and 384: 5.14. FIFTH PROPERTY OF EUCLIDEAN M
- 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
- Page 417 and 418: 6.7. VECTORIZATION & PROJECTION INT
- Page 419 and 420: 6.8. DUAL EDM CONE 419When the Fins
- Page 421 and 422: 6.8. DUAL EDM CONE 421Proof. First,
372 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXλ(−V T N OV N) j90080070060050040030020010001 2 3 4 5 6 7 8 9 10jFigure 91: Largest ten eigenvalues of −VN TOV N for map of USA, sorted bynonincreasing value. In the code (F.3.2), we normalize O by (N(N −1)/2) 2 .Any process of reconstruction that leaves comparative distanceinformation intact is called ordinal multidimensional scaling or isotonicreconstruction. Beyond rotation, reflection, and translation error, (5.5)list reconstruction by isotonic reconstruction is subject to error in absolutescale (dilation) and distance ratio. Yet Borg & Groenen argue: [39,2.2]reconstruction from complete comparative distance information for a largenumber of points is as highly constrained as reconstruction from an EDM;the larger the number, the better.5.13.2.1 Isotonic map of the USATo 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 (943), 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 91: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(948)
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