v2009.01.01 - Convex Optimization

5.12. LIST RECONSTRUCTION 425
where √ ΛQ T pQ p
√
Λ ∆ = Λ = √ Λ √ Λ and where Q p ∈ R n×N−1 is unknown as
is its dimension n . Rotation/reflection is accounted for by Q p yet only its
first r columns are necessarily orthonormal. 5.52 Assuming membership to
the unit simplex y ∈ S (992), then point p = X √ 2V N y = Q p
√
ΛQ T y in R n
belongs to the translated polyhedron
P − x 1 (1036)
whose generating list constitutes the columns of (934)
[ √ ] [ √ ]
0 X 2VN = 0 Q p ΛQ
T
∈ R n×N
= [0 x 2 −x 1 x 3 −x 1 · · · x N −x 1 ]
(1037)
The scaled auxiliary matrix V N represents that translation. A simple choice
for Q p has n set to N − 1; id est, Q p = I . Ideally, each member of the
generating list has at most r nonzero entries; r being, affine dimension
rankV T NDV N = rankQ p
√
ΛQ T = rank Λ = r (1038)
Each member then has at least N −1 − r zeros in its higher-dimensional
coordinates because r ≤ N −1. (946) To truncate those zeros, choose n
equal to affine dimension which is the smallest n possible because XV N has
rank r ≤ n (942). 5.53 In that case, the simplest choice for Q p is [ I 0 ]
having dimensions r ×N −1.
We may wish to verify the list (1037) 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: (798)
5.52 Recall r signifies affine dimension. Q p is not necessarily an orthogonal matrix. Q p is
constrained such that only its first r columns are necessarily orthonormal because there
are only r nonzero eigenvalues in Λ when −VN TDV N has rank r (5.7.1.1). Remaining
columns of Q p are arbitrary.
⎡
⎤
⎡
q T1 ⎤
λ 1 0
. .. 5.53 If we write Q T = ⎣
. .. ⎦ as rowwise eigenvectors, Λ = ⎢ λr ⎥
qN−1
T ⎣ 0 ... ⎦
0 0
in terms of eigenvalues, and Q p = [ ]
q p1 · · · q pN−1 as column vectors, then
√
Q p Λ Q T ∑
= r √
λi q pi qi
T is a sum of r linearly independent rank-one matrices (B.1.1).
i=1
Hence the summation has rank r .