v2006.03.09 - Convex Optimization

4.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 2514.6 Injectivity of D & unique reconstructionInjectivity implies uniqueness of isometric reconstruction; hence, we endeavorto demonstrate it.EDM operators list-form D(X) (460), Gram-form D(G) (472), andinner-product form D(Θ) (515) are many-to-one surjections (4.5) onto thesame range; the EDM cone (5): (confer (473) (544))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= { (538)D(Θ) : R N−1×N−1 → S N h | Θ ∈ R N−1×N−1}where (4.5.1.1)S N⊥c = {u1 T + 1u T | u∈ R N } ⊆ S N (1563)4.6.1 Gram-form bijectivityBecause linear Gram-form EDM operatorD(G) = δ(G)1 T + 1δ(G) T − 2G (472)has no nullspace [47,A.1] on the geometric center subspace 4.17 (E.7.2.0.2)S N c∆= {G∈ S N | G1 = 0} (1561)= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}= {V Y V | Y ∈ S N } ⊂ S N (1562)≡ {V N AV T N | A ∈ SN−1 }(539)then D(G) on that subspace is injective.4.17 The equivalence ≡ in (539) 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 .