v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
448 CHAPTER 7. PROXIMITY PROBLEMS7.1.2 Generic problemPrior to determination of D ⋆ , analytical solution (1118) to Problem 1 isequivalent to solution of a generic rank-constrained projection problem; aEuclidean projection on a rank ρ subset of a PSD cone (on a generallynonconvex subset of the PSD cone boundary ∂S N−1+ when ρ
7.1. FIRST PREVALENT PROBLEM: 449dimension ρ ; 7.7 called rank ρ subset: (185) (219)S N−1+ \ S N−1+ (ρ + 1) = {X ∈ S N−1+ | rankX ≤ ρ} (224)7.1.3 Choice of spectral coneSpectral projection substitutes projection on a polyhedral cone, containinga complete set of eigenspectra, in place of projection on a convex set ofdiagonalizable matrices; e.g., (1138). In this section we develop a method ofspectral projection for constraining rank of positive semidefinite matrices ina proximity problem like (1125). We will see why an orthant turns out to bethe best choice of spectral cone, and why presorting is critical.Define a nonlinear permutation operator π(x) : R n → R n that sorts itsvector argument x into nonincreasing order.7.1.3.0.1 Definition. Spectral projection.Let R be an orthogonal matrix and Λ a nonincreasingly ordered diagonalmatrix of eigenvalues. Spectral projection means unique minimum-distanceprojection of a rotated (R ,B.5.4) nonincreasingly ordered (π) vector (δ)of eigenvaluesπ ( δ(R T ΛR) ) (1127)on a polyhedral cone containing all eigenspectra corresponding to a rank ρsubset of a positive semidefinite cone (2.9.2.1) or the EDM cone (inCayley-Menger form,5.11.2.3).△In the simplest and most common case, projection on a positivesemidefinite cone, orthogonal matrix R equals I (7.1.4.0.1) and diagonalmatrix Λ is ordered during diagonalization (A.5.2). Then spectralprojection simply means projection of δ(Λ) on a subset of the nonnegativeorthant, as we shall now ascertain:It is curious how nonconvex Problem 1 has such a simple analyticalsolution (1118). Although solution to generic problem (1125) is known since1936 [84], Trosset [262,2] first observed its equivalence in 1997 to projection7.7 Recall: affine dimension is a lower bound on embedding (2.3.1), equal to dimensionof the smallest affine set in which points from a list X corresponding to an EDM D canbe embedded.
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- 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
- Page 425 and 426: 6.8. DUAL EDM CONE 425whose veracit
- Page 427 and 428: 6.8. DUAL EDM CONE 4276.8.1.3.1 Exe
- Page 429 and 430: 6.8. DUAL EDM CONE 429has dual affi
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- Page 441 and 442: 441HS N h0EDM NK = S N h ∩ R N×N
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- Page 480 and 481: 480 APPENDIX A. LINEAR ALGEBRAA.1.1
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448 CHAPTER 7. PROXIMITY PROBLEMS7.1.2 Generic problemPrior to determination of D ⋆ , analytical solution (1118) to Problem 1 isequivalent to solution of a generic rank-constrained projection problem; aEuclidean projection on a rank ρ subset of a PSD cone (on a generallynonconvex subset of the PSD cone boundary ∂S N−1+ when ρ
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