v2009.01.01 - Convex Optimization

5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 387
5.6 Injectivity of D & unique reconstruction
Injectivity implies uniqueness of isometric reconstruction; hence, we endeavor
to demonstrate it.
EDM operators list-form D(X) (798), Gram-form D(G) (810), and
inner-product form D(Θ) (863) are many-to-one surjections (5.5) onto the
same range; the EDM cone (6): (confer (811) (900))
EDM N = { D(X) : R N−1×N → S N h | X ∈ R N−1×N}
= { }
D(G) : S N → S N h | G ∈ S N + − S N⊥
c
= { (894)
D(Θ) : R N−1×N−1 → S N h | Θ ∈ R N−1×N−1}
where (5.5.1.1)
S N⊥
c = {u1 T + 1u T | u∈ R N } ⊆ S N (1876)
5.6.1 Gram-form bijectivity
Because linear Gram-form EDM operator
D(G) = δ(G)1 T + 1δ(G) T − 2G (810)
has no nullspace [75,A.1] on the geometric center subspace 5.24 (E.7.2.0.2)
S N c
∆
= {G∈ S N | G1 = 0} (1874)
= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}
= {V Y V | Y ∈ S N } ⊂ S N (1875)
≡ {V N AV T N | A ∈ SN−1 }
(895)
then D(G) on that subspace is injective.
5.24 The equivalence ≡ in (895) follows from the fact: Given B = V Y V = V N AVN T ∈ SN c
with only matrix A∈ S N−1 unknown, then V † †T
NBV N = A or V † N Y V †T
N = A.