v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
588 CHAPTER 7. PROXIMITY PROBLEMS∥ ∥∥∥ N+12∑minimize Υ ⋆R ij , r iiiR ii − Λ∥ + 〈G, W 〉i=1Fsubject to trR ii = 1, i=1... N+1trR ij ⎡= 0,⎤i
7.4. CONCLUSION 589filtering, multidimensional scaling or principal component analysis (5.12),medical imaging (Figure 107), molecular conformation (Figure 127), andsensor-network localization or wireless location (Figure 85).There has been little progress in spectral projection since the discovery byEckart & Young in 1936 [129] leading to a formula for projection on a rank ρsubset of a positive semidefinite cone (2.9.2.1). [146] The only closed-formspectral method presently available for solving proximity problems, having aconstraint on rank, is based on their discovery (Problem 1,7.1,5.13).One popular recourse is intentional misapplication of Eckart & Young’sresult by introducing spectral projection on a positive semidefinite coneinto Problem 3 via D(G) (903), for example. [73] Since Problem 3instead demands spectral projection on the EDM cone, any solutionacquired that way is necessarily suboptimal.A second recourse is problem redesign: A presupposition to allproximity problems in this chapter is that matrix H is given.We considered H having various properties such as nonnegativity,symmetry, hollowness, or lack thereof. It was assumed that if H didnot already belong to the EDM cone, then we wanted an EDM closestto H in some sense; id est, input-matrix H was assumed corruptedsomehow. For practical problems, it withstands reason that such aproximity problem could instead be reformulated so that some or allentries of H were unknown but bounded above and below by knownlimits; the norm objective is thereby eliminated as in the developmentbeginning on page 323. That particular redesign (the art, p.8), in termsof the Gram-matrix bridge between point-list X and EDM D , at onceencompasses proximity and completion problems.A third recourse is to apply the method of convex iteration just likewe did in7.2.2.7.1. This technique is applicable to any semidefiniteproblem requiring a rank constraint; it places a regularization term inthe objective that enforces the rank constraint.
- Page 537 and 538: 6.8. DUAL EDM CONE 537Elegance of t
- Page 539 and 540: 6.8. DUAL EDM CONE 5396.8.1.5 Affin
- Page 541 and 542: 6.8. DUAL EDM CONE 5416.8.1.7 Schoe
- Page 543 and 544: 6.8. DUAL EDM CONE 5430dvec rel ∂
- Page 545 and 546: 6.10. POSTSCRIPT 5456.10 Postscript
- Page 547 and 548: Chapter 7Proximity problemsIn the
- Page 549 and 550: 549project on the subspace, then pr
- Page 551 and 552: 551HS N h0EDM NK = S N h ∩ R N×N
- Page 553 and 554: 5537.0.3 Problem approachProblems t
- Page 555 and 556: 7.1. FIRST PREVALENT PROBLEM: 555fi
- Page 557 and 558: 7.1. FIRST PREVALENT PROBLEM: 5577.
- Page 559 and 560: 7.1. FIRST PREVALENT PROBLEM: 559di
- Page 561 and 562: 7.1. FIRST PREVALENT PROBLEM: 5617.
- Page 563 and 564: 7.1. FIRST PREVALENT PROBLEM: 563wh
- Page 565 and 566: 7.1. FIRST PREVALENT PROBLEM: 565Th
- Page 567 and 568: 7.2. SECOND PREVALENT PROBLEM: 567O
- Page 569 and 570: 7.2. SECOND PREVALENT PROBLEM: 569S
- Page 571 and 572: 7.2. SECOND PREVALENT PROBLEM: 571r
- Page 573 and 574: 7.2. SECOND PREVALENT PROBLEM: 573w
- Page 575 and 576: 7.2. SECOND PREVALENT PROBLEM: 5757
- Page 577 and 578: 7.2. SECOND PREVALENT PROBLEM: 577a
- Page 579 and 580: 7.3. THIRD PREVALENT PROBLEM: 579is
- Page 581 and 582: 7.3. THIRD PREVALENT PROBLEM: 581We
- Page 583 and 584: 7.3. THIRD PREVALENT PROBLEM: 583su
- Page 585 and 586: 7.3. THIRD PREVALENT PROBLEM: 585Gi
- Page 587: 7.3. THIRD PREVALENT PROBLEM: 587Op
- Page 591 and 592: Appendix ALinear algebraA.1 Main-di
- Page 593 and 594: A.1. MAIN-DIAGONAL δ OPERATOR, λ
- Page 595 and 596: A.1. MAIN-DIAGONAL δ OPERATOR, λ
- Page 597 and 598: A.2. SEMIDEFINITENESS: DOMAIN OF TE
- Page 599 and 600: A.3. PROPER STATEMENTS 599(AB) T
- Page 601 and 602: A.3. PROPER STATEMENTS 601A.3.1Semi
- Page 603 and 604: A.3. PROPER STATEMENTS 603For A dia
- Page 605 and 606: A.3. PROPER STATEMENTS 605Diagonali
- Page 607 and 608: A.3. PROPER STATEMENTS 607For A,B
- Page 609 and 610: A.3. PROPER STATEMENTS 609When B is
- Page 611 and 612: A.4. SCHUR COMPLEMENT 611A.4 Schur
- Page 613 and 614: A.4. SCHUR COMPLEMENT 613A.4.0.0.3
- Page 615 and 616: A.4. SCHUR COMPLEMENT 615From Corol
- Page 617 and 618: A.5. EIGENVALUE DECOMPOSITION 617wh
- Page 619 and 620: A.5. EIGENVALUE DECOMPOSITION 619A.
- Page 621 and 622: A.6. SINGULAR VALUE DECOMPOSITION,
- Page 623 and 624: A.6. SINGULAR VALUE DECOMPOSITION,
- Page 625 and 626: A.7. ZEROS 625A.6.5SVD of symmetric
- Page 627 and 628: A.7. ZEROS 627(Transpose.)Likewise,
- Page 629 and 630: A.7. ZEROS 629For X,A∈ S M +[34,2
- Page 631 and 632: A.7. ZEROS 631A.7.5.0.1 Proposition
- Page 633 and 634: Appendix BSimple matricesMathematic
- Page 635 and 636: B.1. RANK-ONE MATRIX (DYAD) 635R(v)
- Page 637 and 638: B.1. RANK-ONE MATRIX (DYAD) 637B.1.
588 CHAPTER 7. PROXIMITY PROBLEMS∥ ∥∥∥ N+12∑minimize Υ ⋆R ij , r iiiR ii − Λ∥ + 〈G, W 〉i=1Fsubject to trR ii = 1, i=1... N+1trR ij ⎡= 0,⎤i