v2010.10.26 - Convex Optimization

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342 CHAPTER 4. SEMIDEFINITE PROGRAMMINGWe also assume that the gap’s beginning and ending in time are preciselylocalized to within a sample; id est, index l locates the last sample prior tothe gap’s onset, while index n−l+1 locates the first sample subsequent tothe gap: for rectangularly windowed received signal g possessing a time-gaploss and additive noise η ∈ R n⎡⎤s 1:l + η 1:lg = ⎣ η l+1:n−l⎦∈ R n (771)+ η n−l+1:ns n−l+1:nThe window is thereby centered on the gap and short enough so that the DCTspectrum of signal s can be assumed static over the window’s duration n .Signal to noise ratio within this window is defined[ ]∥ s 1:l ∥∥∥∥ s n−l+1:nSNR 20 log(772)‖η‖In absence of noise, knowing the signal DCT basis and having agood estimate of basis coefficient cardinality makes perfectly reconstructinggap-loss easy: it amounts to solving a linear system of equations andrequires little or no optimization; with caveat, number of equations exceedscardinality of signal representation (roughly l ≥ k) with respect to DCTbasis.But addition of a significant amount of noise η increases level of difficultydramatically; a 1-norm based method of reducing cardinality, for example,almost always returns DCT basis coefficients numbering in excess of minimalcardinality. We speculate that is because signal cardinality 2l becomes thepredominant cardinality. DCT basis coefficient cardinality is an explicitconstraint to the optimization problem we shall pose: In presence of noise,constraints equating reconstructed signal f to received signal g are notpossible. We can instead formulate the dropout recovery problem as a bestapproximation:∥[∥∥∥minimizex∈R nsubject to f = Ψxx ≽ 0card x ≤ k]∥f 1:l − g 1:l ∥∥∥f n−l+1:n − g n−l+1:n(773)
4.5. CONSTRAINING CARDINALITY 343flatline and g0.250.20.15s + η0.10.050−0.05ηs + η(a)−0.1−0.15−0.2dropout (s = 0)−0.250 100 200 300 400 500f and g0.250.20.15f0.10.050(b)−0.05−0.1−0.15−0.2−0.250 100 200 300 400 500Figure 101: (a) Signal dropout in signal s corrupted by noise η(SNR =10dB, g = s + η). Flatline indicates duration of signal dropout.(b) Reconstructed signal f (red) overlaid with corrupted signal g .
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DATTORROCONVEXOPTIMIZATION&EUCLIDEA
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Convex Optimization&Euclidean Dista
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for Jennie Columba♦Antonio♦♦&
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PreludeThe constant demands of my d
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Convex Optimization&Euclidean Dista
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CONVEX OPTIMIZATION & EUCLIDEAN DIS
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List of Figures1 Overview 211 Orion
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LIST OF FIGURES 1562 Shrouded polyh
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LIST OF FIGURES 17130 Elliptope E 3
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Chapter 1OverviewConvex Optimizatio
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ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
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25Figure 4: This coarsely discretiz
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(biorthogonal expansion) is examine
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29cardinality Boolean solution to a
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31Figure 8: Robotic vehicles in con
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an elaborate exposition offering in
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Chapter 2Convex geometryConvexity h
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2.1. CONVEX SET 372.1.2 linear inde
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2.1. CONVEX SET 392.1.6 empty set v
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2.1. CONVEX SET 41(a)R(b)R 2(c)R 3F
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2.1. CONVEX SET 43where Q∈ R 3×3
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2.1. CONVEX SET 45Now let’s move
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2.1. CONVEX SET 47By additive inver
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2.1. CONVEX SET 49R nR mR(A T )x px
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 61Figure 20: Convex hull
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2.3. HULLS 63Aaffine hull (drawn tr
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2.3. HULLS 65subset of the affine h
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2.3. HULLS 672.3.2.0.2 Example. Nuc
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2.3. HULLS 692.3.2.0.3 Exercise. Co
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2.3. HULLS 71Figure 24: A simplicia
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2.4. HALFSPACE, HYPERPLANE 73H + =
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2.4. HALFSPACE, HYPERPLANE 7511−1
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2.4. HALFSPACE, HYPERPLANE 772.4.2.
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2.4. HALFSPACE, HYPERPLANE 81tradit
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2.5. SUBSPACE REPRESENTATIONS 852.5
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2.5. SUBSPACE REPRESENTATIONS 87(Ex
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2.5. SUBSPACE REPRESENTATIONS 89(a)
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2.5. SUBSPACE REPRESENTATIONS 91are
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2.6. EXTREME, EXPOSED 93A one-dimen
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2.6. EXTREME, EXPOSED 952.6.1.1 Den
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2.7. CONES 97X(a)00(b)XFigure 33: (
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2.7. CONES 99XXFigure 37: Truncated
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2.7. CONES 101Figure 39: Not a cone
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2.7. CONES 103cone that is a halfli
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2.7. CONES 105A pointed closed conv
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2.8. CONE BOUNDARY 107That means th
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2.8. CONE BOUNDARY 1092.8.1.1 extre
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2.8. CONE BOUNDARY 1112.8.2 Exposed
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 141
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2.12. CONVEX POLYHEDRA 147all dimen
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2.12. CONVEX POLYHEDRA 149convex po
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2.12. CONVEX POLYHEDRA 1512.12.2.2
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2.13. DUAL CONE & GENERALIZED INEQU
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Chapter 3Geometry of convex functio
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3.1. CONVEX FUNCTION 221f 1 (x)f 2
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3.1. CONVEX FUNCTION 223Rf(b)f(X
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3.2. PRACTICAL NORM FUNCTIONS, ABSO
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3.3. INVERTED FUNCTIONS AND ROOTS 2
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3.4. AFFINE FUNCTION 237rather]x >
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3.4. AFFINE FUNCTION 239f(z)Az 2z 1
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3.5. EPIGRAPH, SUBLEVEL SET 241{a T
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3.5. EPIGRAPH, SUBLEVEL SET 243Subl
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3.5. EPIGRAPH, SUBLEVEL SET 245wher
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3.5. EPIGRAPH, SUBLEVEL SET 247part
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3.5. EPIGRAPH, SUBLEVEL SET 249that
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3.6. GRADIENT 251respect to its vec
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3.6. GRADIENT 253Invertibility is g
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3.6. GRADIENT 2553.6.1.0.2 Theorem.
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3.6. GRADIENT 257f(Y )[ ∇f(X)−1
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3.6. GRADIENT 259αβα ≥ β ≥
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3.6. GRADIENT 2613.6.4 second-order
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3.7. CONVEX MATRIX-VALUED FUNCTION
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3.8. QUASICONVEX 269exponential alw
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3.9. SALIENT PROPERTIES 2713.8.0.0.
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Chapter 4Semidefinite programmingPr
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4.1. CONIC PROBLEM 275(confer p.162
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4.1. CONIC PROBLEM 277PCsemidefinit
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4.1. CONIC PROBLEM 279is the affine
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4.1. CONIC PROBLEM 281faces of S 3
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4.1. CONIC PROBLEM 2834.1.2.3 Previ
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4.2. FRAMEWORK 285Semidefinite Fark
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4.2. FRAMEWORK 287On the other hand
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4.2. FRAMEWORK 2894.2.2.1 Dual prob
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Page 295 and 296: 4.2. FRAMEWORK 295minimize 1 TˆxX
Page 297 and 298: 4.2. FRAMEWORK 297asminimize ‖ỹ
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4.8. CONVEX ITERATION RANK-1 393the
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Chapter 5Euclidean Distance MatrixT
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5.2. FIRST METRIC PROPERTIES 397to
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5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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5.4. EDM DEFINITION 403The collecti
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5.4. EDM DEFINITION 4055.4.2 Gram-f
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5.4. EDM DEFINITION 407We provide a
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5.4. EDM DEFINITION 4095.4.2.3.1 Ex
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5.4. EDM DEFINITION 411is the fact:
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5.4. EDM DEFINITION 413Figure 120:
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5.4. EDM DEFINITION 415is found fro
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5.4. EDM DEFINITION 417one less dim
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5.4. EDM DEFINITION 419equality con
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5.4. EDM DEFINITION 421How much dis
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5.4. EDM DEFINITION 423105ˇx 4ˇx
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5.4. EDM DEFINITION 425now implicit
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5.4. EDM DEFINITION 427by translate
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5.4. EDM DEFINITION 429Crippen & Ha
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5.4. EDM DEFINITION 431where ([√t
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5.4. EDM DEFINITION 433because (A.3
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5.5. INVARIANCE 4355.5.1.0.1 Exampl
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5.5. INVARIANCE 437x 2 x 2x 3 x 1 x
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 4455.
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5.7. EMBEDDING IN AFFINE HULL 447Fo
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5.7. EMBEDDING IN AFFINE HULL 4495.
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5.8. EUCLIDEAN METRIC VERSUS MATRIX
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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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5.10. EDM-ENTRY COMPOSITION 465of a
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5.10. EDM-ENTRY COMPOSITION 467Then
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5.11. EDM INDEFINITENESS 4695.11.1
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5.11. EDM INDEFINITENESS 471we have
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5.11. EDM INDEFINITENESS 473So beca
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5.11. EDM INDEFINITENESS 475holds o
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5.12. LIST RECONSTRUCTION 477where
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5.12. LIST RECONSTRUCTION 479(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 481Wi
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5.13. RECONSTRUCTION EXAMPLES 483d
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5.13. RECONSTRUCTION EXAMPLES 485Th
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6Cone of distance matricesF
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6.1. DEFINING EDM CONE 4996.1 Defin
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6.2. POLYHEDRAL BOUNDS 501This cone
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6.4. EDM DEFINITION IN 11 T 503That
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6.4. EDM DEFINITION IN 11 T 505N(1
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6.4. EDM DEFINITION IN 11 T 507Then
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6.4. EDM DEFINITION IN 11 T 5096.4.
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6.5. CORRESPONDENCE TO PSD CONE S N
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6.6. VECTORIZATION & PROJECTION INT
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6.7. A GEOMETRY OF COMPLETION 523(b
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6.7. A GEOMETRY OF COMPLETION 525[3
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6.7. A GEOMETRY OF COMPLETION 527wh
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6.8. DUAL EDM CONE 529to the geomet
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6.8. DUAL EDM CONE 531Proof. First,
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6.8. DUAL EDM CONE 533EDM 2 = S 2 h
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6.8. DUAL EDM CONE 535therefore the
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6.8. DUAL EDM CONE 537Elegance of t
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6.8. DUAL EDM CONE 5396.8.1.5 Affin
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6.8. DUAL EDM CONE 5416.8.1.7 Schoe
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6.8. DUAL EDM CONE 5430dvec rel ∂
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6.10. POSTSCRIPT 5456.10 Postscript
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Chapter 7Proximity problemsIn the
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549project on the subspace, then pr
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551HS N h0EDM NK = S N h ∩ R N×N
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5537.0.3 Problem approachProblems t
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7.1. FIRST PREVALENT PROBLEM: 555fi
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7.1. FIRST PREVALENT PROBLEM: 5577.
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7.1. FIRST PREVALENT PROBLEM: 559di
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7.1. FIRST PREVALENT PROBLEM: 5617.
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7.1. FIRST PREVALENT PROBLEM: 563wh
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7.1. FIRST PREVALENT PROBLEM: 565Th
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7.2. SECOND PREVALENT PROBLEM: 567O
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7.2. SECOND PREVALENT PROBLEM: 569S
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7.2. SECOND PREVALENT PROBLEM: 571r
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7.2. SECOND PREVALENT PROBLEM: 573w
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7.2. SECOND PREVALENT PROBLEM: 5757
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7.2. SECOND PREVALENT PROBLEM: 577a
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7.3. THIRD PREVALENT PROBLEM: 579is
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7.3. THIRD PREVALENT PROBLEM: 581We
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7.3. THIRD PREVALENT PROBLEM: 583su
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7.3. THIRD PREVALENT PROBLEM: 585Gi
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7.3. THIRD PREVALENT PROBLEM: 587Op
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7.4. CONCLUSION 589filtering, multi
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Appendix ALinear algebraA.1 Main-di
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.3. PROPER STATEMENTS 599(AB) T
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A.3. PROPER STATEMENTS 601A.3.1Semi
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A.3. PROPER STATEMENTS 603For A dia
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A.3. PROPER STATEMENTS 605Diagonali
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A.3. PROPER STATEMENTS 607For A,B
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A.3. PROPER STATEMENTS 609When B is
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A.4. SCHUR COMPLEMENT 611A.4 Schur
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A.4. SCHUR COMPLEMENT 613A.4.0.0.3
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A.4. SCHUR COMPLEMENT 615From Corol
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A.5. EIGENVALUE DECOMPOSITION 617wh
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A.5. EIGENVALUE DECOMPOSITION 619A.
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.7. ZEROS 625A.6.5SVD of symmetric
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A.7. ZEROS 627(Transpose.)Likewise,
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A.7. ZEROS 629For X,A∈ S M +[34,2
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A.7. ZEROS 631A.7.5.0.1 Proposition
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Appendix BSimple matricesMathematic
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B.1. RANK-ONE MATRIX (DYAD) 635R(v)
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B.1. RANK-ONE MATRIX (DYAD) 637B.1.
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B.2. DOUBLET 639R([u v ])R(Π)= R([
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B.3. ELEMENTARY MATRIX 641has N −
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B.4. AUXILIARY V -MATRICES 643is an
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B.4. AUXILIARY V -MATRICES 64514. [
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B.5. ORTHOGONAL MATRIX 647Given X
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B.5. ORTHOGONAL MATRIX 649Figure 15
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B.5. ORTHOGONAL MATRIX 651which is
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Appendix CSome analytical optimal r
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C.2. TRACE, SINGULAR AND EIGEN VALU
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C.3. ORTHOGONAL PROCRUSTES PROBLEM
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Appendix DMatrix calculusFrom too m
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D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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D.2. TABLES OF GRADIENTS AND DERIVA
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Appendix EProjectionFor any A∈ R
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701U T = U † for orthonormal (inc
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E.1. IDEMPOTENT MATRICES 703where A
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E.1. IDEMPOTENT MATRICES 705order,
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E.1. IDEMPOTENT MATRICES 707are lin
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.4. ALGEBRA OF PROJECTION ON AFFIN
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E.5. PROJECTION EXAMPLES 717a ∗ 2
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E.5. PROJECTION EXAMPLES 719where Y
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E.5. PROJECTION EXAMPLES 721(B.4.2)
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E.6. VECTORIZATION INTERPRETATION,
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E.7. PROJECTION ON MATRIX SUBSPACES
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E.7. PROJECTION ON MATRIX SUBSPACES
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E.8. RANGE/ROWSPACE INTERPRETATION
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E.9. PROJECTION ON CONVEX SET 735As
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E.9. PROJECTION ON CONVEX SET 737Wi
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E.9. PROJECTION ON CONVEX SET 739R(
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E.9. PROJECTION ON CONVEX SET 741E.
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E.9. PROJECTION ON CONVEX SET 743E.
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E.9. PROJECTION ON CONVEX SET 745Un
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E.9. PROJECTION ON CONVEX SET 747ac
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E.10. ALTERNATING PROJECTION 749bC
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E.10. ALTERNATING PROJECTION 7510
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E.10. ALTERNATING PROJECTION 753E.1
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E.10. ALTERNATING PROJECTION 755y 2
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E.10. ALTERNATING PROJECTION 757Def
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E.10. ALTERNATING PROJECTION 759Dis
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E.10. ALTERNATING PROJECTION 761mat
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E.10. ALTERNATING PROJECTION 763K
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Appendix FNotation and a few defini
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771A ij or A(i, j) , ij th entry of
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773⊞orthogonal vector sum of sets
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775x +vector x whose negative entri
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777X point list ((76) having cardin
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779SDPSVDSNRdBEDMS n 1S n hS n⊥hS
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781vectorentrycubixquartixfeasible
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783Oorder of magnitude information
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785cofmatrix of cofactors correspon
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Bibliography[1] Edwin A. Abbott. Fl
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BIBLIOGRAPHY 789[23] Dror Baron, Mi
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BIBLIOGRAPHY 791[49] Leonard M. Blu
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BIBLIOGRAPHY 793[74] Yves Chabrilla
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BIBLIOGRAPHY 795[102] Etienne de Kl
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BIBLIOGRAPHY 797[129] Carl Eckart a
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BIBLIOGRAPHY 799[154] James Gleik.
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BIBLIOGRAPHY 801[182] Johan Håstad
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BIBLIOGRAPHY 803[212] Viren Jain an
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BIBLIOGRAPHY 805[237] Monique Laure
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BIBLIOGRAPHY 807[265] Sunderarajan
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BIBLIOGRAPHY 809Notes in Computer S
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BIBLIOGRAPHY 811[319] Anthony Man-C
Page 813 and 814:
BIBLIOGRAPHY 813[346] Pham Dinh Tao
Page 815 and 816:
BIBLIOGRAPHY 815[375] Bernard Widro
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Index∅, see empty set0-norm, 229,
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INDEX 819bees, 30, 413Bellman, 276b
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INDEX 821circular, 130construction,
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INDEX 823measure, 295convex, 21, 36
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INDEX 825duality, 160, 410gap, 161,
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INDEX 827positive semidefinite, 284
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INDEX 829point, 39, 62positive semi
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INDEX 831isomorphic, 52, 56, 59, 77
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INDEX 833nonsymmetric, 703range, 70
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INDEX 835Muller, 624multidimensiona
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INDEX 837projection on, 739, 745tra
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INDEX 839pseudoinverse, 701trace, 5
Page 841 and 842:
INDEX 841coordinate system, 156line
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INDEX 843nonconvex, 40, 97-101nulls
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INDEX 845faces, 106intersection, 10
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INDEX 847projection, 517symmetric,
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849
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Convex Optimization & Euclidean Dis
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v2010.10.26 - Convex Optimization

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