v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
558 CHAPTER 7. PROXIMITY PROBLEMS7.1.2 Generic problemPrior to determination of D ⋆ , analytical solution (1316) to Problem 1 isequivalent to solution of a generic rank-constrained projection problem:Given desired affine dimension ρ andA −V T NHV N =N−1∑i=1λ i v i v T i ∈ S N−1 (1317)Euclidean 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: 559dimension ρ ; 7.9 called rank ρ subset: (260)S N−1+ \ S N−1+ (ρ + 1) = {X ∈ S N−1+ | rankX ≤ ρ} (216)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., (1336). In this section we develop a method ofspectral projection for constraining rank of positive semidefinite matrices ina proximity problem like (1323). 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) ) (1325)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.5.1). 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 (1316). Although solution to generic problem (1323) is well knownsince 1936 [129], its equivalence was observed in 1997 [353,2] to projection7.9 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.
- Page 507 and 508: 6.4. EDM DEFINITION IN 11 T 507Then
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- Page 529 and 530: 6.8. DUAL EDM CONE 529to the geomet
- Page 531 and 532: 6.8. DUAL EDM CONE 531Proof. First,
- Page 533 and 534: 6.8. DUAL EDM CONE 533EDM 2 = S 2 h
- Page 535 and 536: 6.8. DUAL EDM CONE 535therefore the
- 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
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- Page 543 and 544: 6.8. DUAL EDM CONE 5430dvec rel ∂
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- 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
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- Page 589 and 590: 7.4. CONCLUSION 589filtering, multi
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- Page 607 and 608: A.3. PROPER STATEMENTS 607For A,B
558 CHAPTER 7. PROXIMITY PROBLEMS7.1.2 Generic problemPrior to determination of D ⋆ , analytical solution (1316) to Problem 1 isequivalent to solution of a generic rank-constrained projection problem:Given desired affine dimension ρ andA −V T NHV N =N−1∑i=1λ i v i v T i ∈ S N−1 (1317)Euclidean projection on a rank ρ subset of a PSD cone (on a generallynonconvex subset of the PSD cone boundary ∂S N−1+ when ρ