v2007.09.13 - Convex Optimization

5.12. LIST RECONSTRUCTION 367where √ ΛQ T pQ p√Λ ∆ = Λ = √ Λ √ Λ and where Q p ∈ R n×N−1 is unknown asis its dimension n . Rotation/reflection is accounted for by Q p yet only itsfirst r columns are necessarily orthonormal. 5.51 Assuming membership tothe unit simplex y ∈ S (890), then point p = X √ 2V N y = Q p√ΛQ T y in R nbelongs to the translated polyhedronP − x 1 (934)whose generating list constitutes the columns of (834)[ √ ] [ √ ]0 X 2VN = 0 Q p ΛQT∈ R n×N= [0 x 2 −x 1 x 3 −x 1 · · · x N −x 1 ](935)The scaled auxiliary matrix V N represents that translation. A simple choicefor Q p has n set to N − 1; id est, Q p = I . Ideally, each member of thegenerating list has at most r nonzero entries; r being, affine dimensionrankV T NDV N = rankQ p√ΛQ T = rank Λ = r (936)Each member then has at least N −1 − r zeros in its higher-dimensionalcoordinates because r ≤ N −1. (846) To truncate those zeros, choose nequal to affine dimension which is the smallest n possible because XV N hasrank r ≤ n (842). 5.52 In that case, the simplest choice for Q p is [ I 0 ]having dimensions r ×N −1.We may wish to verify the list (935) found from the diagonalization of−VN TDV N . Because of rotation/reflection and translation invariance (5.5),EDM D can be uniquely made from that list by calculating: (705)5.51 Recall r signifies affine dimension. Q p is not necessarily an orthogonal matrix. Q p isconstrained such that only its first r columns are necessarily orthonormal because thereare only r nonzero eigenvalues in Λ when −VN TDV N has rank r (5.7.1.1). Remainingcolumns of Q p are arbitrary.⎡⎤⎡q T1 ⎤λ 1 0. .. 5.52 If we write Q T = ⎣. .. ⎦ as rowwise eigenvectors, Λ = ⎢ λr ⎥qN−1T ⎣ 0 ... ⎦0 0in terms of eigenvalues, and Q p = [ ]q p1 · · · q pN−1 as column vectors, then√Q p Λ Q T ∑= r √λi q pi qiT is a sum of r linearly independent rank-one matrices (B.1.1).i=1Hence the summation has rank r .