v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
374 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXThat sort constraint demands: any optimal solution D ⋆ must possess theknown comparative distance relationship that produces the original ordinaldistance data O (947). 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 (951). This suggeststhe simple reconstruction techniques of5.12 are robust to a significantamount of noise.5.13.2.2 Isotonic solution with sort constraintBecause problems involving rank are generally difficult, we will partition(950) into two problems we know how to solve and then alternate theirsolution until convergence:minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3D ∈ EDM Nminimize ‖σ − Πd‖σsubject to σ ∈ K M+(a)(b)(952)where the sort-index matrix O (a given constant in (a)) becomes an implicitvectorized variable o i solving the i th instance of (952b)o i ∆ = Π T σ ⋆ = 1 √2dvecO i ∈ R N(N−1)/2 , i∈{1, 2, 3...} (953)As mentioned in discussion of relaxed problem (951), a closed-formsolution to problem (952a) exists. Only the first iteration of (952a) sees theoriginal sort-index matrix O whose entries are nonnegative whole numbers;id est, O 0 =O∈ S N h ∩ R N×N+ (947). Subsequent iterations i take the previoussolution of (952b) as inputO i = dvec −1 ( √ 2o i ) ∈ S N (954)real successors to the sort-index matrix O .New problem (952b) finds the unique minimum-distance projection ofΠd on the monotone nonnegative cone K M+ . By definingY †T ∆ = [e 1 − e 2 e 2 −e 3 e 3 −e 4 · · · e m ] ∈ R m×m (371)
5.13. RECONSTRUCTION EXAMPLES 375where m=N(N ∆ −1)/2, we may rewrite (952b) as an equivalent quadraticprogram; a convex optimization problem [46,4] in terms of thehalfspace-description of K M+ :minimize (σ − Πd) T (σ − Πd)σsubject to Y † σ ≽ 0(955)This quadratic program can be converted to a semidefinite program via Schurform (3.1.7.2); we get the equivalent problemminimizet∈R , σsubject tot[tI σ − Πd(σ − Πd) T 1]≽ 0(956)Y † σ ≽ 05.13.2.3 ConvergenceInE.10 we discuss convergence of alternating projection on intersectingconvex sets in a Euclidean vector space; convergence to a point in theirintersection. Here the situation is different for two reasons:Firstly, sets of positive semidefinite matrices having an upper bound onrank are generally not convex. Yet in7.1.4.0.1 we prove (952a) is equivalentto a projection of nonincreasingly ordered eigenvalues on a subset of thenonnegative orthant:minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3D ∈ EDM N≡minimize ‖Υ − Λ‖ FΥ [ ]R3+ (957)subject to δ(Υ) ∈0∆where −VN TDV N =UΥU T ∈ S N−1 and −VN TOV N =QΛQ T ∈ S N−1 areordered diagonalizations (A.5). It so happens: optimal orthogonal U ⋆always equals Q given. Linear operator T(A) = U ⋆T AU ⋆ , acting on squarematrix A , is a bijective isometry because the Frobenius norm is orthogonallyinvariant (40). This isometric isomorphism T thus maps a nonconvexproblem to a convex one that preserves distance.∆
- 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 and 372: 5.13. RECONSTRUCTION EXAMPLES 371D
- Page 373: 5.13. RECONSTRUCTION EXAMPLES 373Th
- 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,
- Page 423 and 424: 6.8. DUAL EDM CONE 423EDM 2 = S 2 h
5.13. RECONSTRUCTION EXAMPLES 375where m=N(N ∆ −1)/2, we may rewrite (952b) as an equivalent quadraticprogram; a convex optimization problem [46,4] in terms of thehalfspace-description of K M+ :minimize (σ − Πd) T (σ − Πd)σsubject to Y † σ ≽ 0(955)This quadratic program can be converted to a semidefinite program via Schurform (3.1.7.2); we get the equivalent problemminimizet∈R , σsubject tot[tI σ − Πd(σ − Πd) T 1]≽ 0(956)Y † σ ≽ 05.13.2.3 ConvergenceInE.10 we discuss convergence of alternating projection on intersectingconvex sets in a Euclidean vector space; convergence to a point in theirintersection. Here the situation is different for two reasons:Firstly, sets of positive semidefinite matrices having an upper bound onrank are generally not convex. Yet in7.1.4.0.1 we prove (952a) is equivalentto a projection of nonincreasingly ordered eigenvalues on a subset of thenonnegative orthant:minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3D ∈ EDM N≡minimize ‖Υ − Λ‖ FΥ [ ]R3+ (957)subject to δ(Υ) ∈0∆where −VN TDV N =UΥU T ∈ S N−1 and −VN TOV N =QΛQ T ∈ S N−1 areordered diagonalizations (A.5). It so happens: optimal orthogonal U ⋆always equals Q given. Linear operator T(A) = U ⋆T AU ⋆ , acting on squarematrix A , is a bijective isometry because the Frobenius norm is orthogonallyinvariant (40). This isometric isomorphism T thus maps a nonconvexproblem to a convex one that preserves distance.∆