v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
348 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.9 Bridge: Convex polyhedra to EDMsThe criteria for the existence of an EDM include, by definition (705) (770),the properties imposed upon its entries d ij by the Euclidean metric. From5.8.1 and5.8.2, we know there is a relationship of matrix criteria to thoseproperties. Here is a snapshot of what we are sure: for i , j , k ∈{1... N}(confer5.2)√dij ≥ 0, i ≠ j√dij = 0, i = j√dij = √ d ji √dij≤ √ d ik + √ d kj , i≠j ≠k⇐−V T N DV N ≽ 0δ(D) = 0D T = D(878)all implied by D ∈ EDM N . In words, these four Euclidean metric propertiesare necessary conditions for D to be a distance matrix. At the moment,we have no converse. As of concern in5.3, we have yet to establishmetric requirements beyond the four Euclidean metric properties that wouldallow D to be certified an EDM or might facilitate polyhedron or listreconstruction from an incomplete EDM. We deal with this problem in5.14.Our present goal is to establish ab initio the necessary and sufficient matrixcriteria that will subsume all the Euclidean metric properties and any furtherrequirements 5.40 for all N >1 (5.8.3); id est,−V T N DV N ≽ 0D ∈ S N h}⇔ D ∈ EDM N (724)or for EDM definition (779),}Ω ≽ 0√δ(d) ≽ 0⇔ D = D(Ω,d) ∈ EDM N (879)5.40 In 1935, Schoenberg [232, (1)] first extolled matrix product −VN TDV N (858)(predicated on symmetry and self-distance) specifically incorporating V N , albeitalgebraically. He showed: nonnegativity −y T VN TDV N y ≥ 0, for all y ∈ R N−1 , is necessaryand sufficient for D to be an EDM. Gower [111,3] remarks how surprising it is that sucha fundamental property of Euclidean geometry was obtained so late.
5.9. BRIDGE: CONVEX POLYHEDRA TO EDMS 349Figure 86: Elliptope E 3 in isometrically isomorphic R 6 (projected on R 3 )is a convex body that appears to possess some kind of symmetry in thisdimension; it resembles a malformed pillow in the shape of a bulgingtetrahedron. Elliptope relative boundary is not smooth and comprises allset members (880) having at least one 0 eigenvalue. [173,2.1] This elliptopehas an infinity of vertices, but there are only four vertices corresponding toa rank-1 matrix. Those yy T , evident in the illustration, have binary vectory ∈ R 3 with entries in {±1}.
- Page 297 and 298: 5.4. EDM DEFINITION 297The collecti
- Page 299 and 300: 5.4. EDM DEFINITION 2995.4.2 Gram-f
- Page 301 and 302: 5.4. EDM DEFINITION 301D ∈ EDM N
- Page 303 and 304: 5.4. EDM DEFINITION 3035.4.2.2.1 Ex
- Page 305 and 306: 5.4. EDM DEFINITION 305ten affine e
- Page 307 and 308: 5.4. EDM DEFINITION 307spheres:Then
- Page 309 and 310: 5.4. EDM DEFINITION 309By eliminati
- Page 311 and 312: 5.4. EDM DEFINITION 311whereΦ ij =
- Page 313 and 314: 5.4. EDM DEFINITION 3135.4.2.2.5 De
- Page 315 and 316: 5.4. EDM DEFINITION 315105ˇx 4ˇx
- 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
- Page 343 and 344: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 345 and 346: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- Page 347: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
- 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 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
- 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
348 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.9 Bridge: <strong>Convex</strong> polyhedra to EDMsThe criteria for the existence of an EDM include, by definition (705) (770),the properties imposed upon its entries d ij by the Euclidean metric. From5.8.1 and5.8.2, we know there is a relationship of matrix criteria to thoseproperties. Here is a snapshot of what we are sure: for i , j , k ∈{1... N}(confer5.2)√dij ≥ 0, i ≠ j√dij = 0, i = j√dij = √ d ji √dij≤ √ d ik + √ d kj , i≠j ≠k⇐−V T N DV N ≽ 0δ(D) = 0D T = D(878)all implied by D ∈ EDM N . In words, these four Euclidean metric propertiesare necessary conditions for D to be a distance matrix. At the moment,we have no converse. As of concern in5.3, we have yet to establishmetric requirements beyond the four Euclidean metric properties that wouldallow D to be certified an EDM or might facilitate polyhedron or listreconstruction from an incomplete EDM. We deal with this problem in5.14.Our present goal is to establish ab initio the necessary and sufficient matrixcriteria that will subsume all the Euclidean metric properties and any furtherrequirements 5.40 for all N >1 (5.8.3); id est,−V T N DV N ≽ 0D ∈ S N h}⇔ D ∈ EDM N (724)or for EDM definition (779),}Ω ≽ 0√δ(d) ≽ 0⇔ D = D(Ω,d) ∈ EDM N (879)5.40 In 1935, Schoenberg [232, (1)] first extolled matrix product −VN TDV N (858)(predicated on symmetry and self-distance) specifically incorporating V N , albeitalgebraically. He showed: nonnegativity −y T VN TDV N y ≥ 0, for all y ∈ R N−1 , is necessaryand sufficient for D to be an EDM. Gower [111,3] remarks how surprising it is that sucha fundamental property of Euclidean geometry was obtained so late.