v2007.11.26 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 465dimension ρ ; 7.7 called rank ρ subset: (186) (220)S N−1+ \ S N−1+ (ρ + 1) = {X ∈ S N−1+ | rankX ≤ ρ} (225)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., (1168). In this section we develop a method ofspectral projection for constraining rank of positive semidefinite matrices ina proximity problem like (1155). 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) ) (1157)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 (1148). Although solution to generic problem (1155) is known since1936 [85], Trosset [266,2] 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.