v2007.09.13 - Convex Optimization

5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 3295.6 Injectivity of D & unique reconstructionInjectivity implies uniqueness of isometric reconstruction; hence, we endeavorto demonstrate it.EDM operators list-form D(X) (705), Gram-form D(G) (717), andinner-product form D(Θ) (770) are many-to-one surjections (5.5) onto thesame range; the EDM cone (6): (confer (718) (800))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= { (794)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 (1763)5.6.1 Gram-form bijectivityBecause linear Gram-form EDM operatorD(G) = δ(G)1 T + 1δ(G) T − 2G (717)has no nullspace [58,A.1] on the geometric center subspace 5.23 (E.7.2.0.2)S N c∆= {G∈ S N | G1 = 0} (1761)= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}= {V Y V | Y ∈ S N } ⊂ S N (1762)≡ {V N AV T N | A ∈ SN−1 }(795)then D(G) on that subspace is injective.5.23 The equivalence ≡ in (795) follows from the fact: Given B = V Y V = V N AVN T ∈ SN cwith only matrix A∈ S N−1 unknown, then V † †TNBV N = A or V † N Y V †TN = A .