v2007.09.13 - Convex Optimization

5.4. EDM DEFINITION 321As before, ascribe position information to the matrixX = [x 1 · · · x N ] ∈ R 3×N (65)and introduce a matrix ℵ holding normals η to planes respecting dihedralangles ϕ :ℵ ∆ = [η 1 · · · η M ] ∈ R 3×M (759)As in the other examples, we preferentially work with Gram matrices Gbecause of the bridge they provide between other variables; we define[ ]Gℵ ZZ T G X∆=[ ℵ T ℵ ℵ T XX T ℵ X T X]=[ ℵTX T ][ ℵ X ] ∈ R N+M×N+M (760)whose rank is 3 by assumption. So our problem’s variables are the twoGram matrices and the matrix Z of inner products. Then measurements ofdistance-square can be expressed as linear constraints on G X as in (758),dihedral angle ϕ measurements can be expressed as linear constraints on G ℵby (734), and the normal-vector conditions can be expressed by vanishinglinear constraints on inner-product matrix Z . Consider three points xlabelled 1, 2, 3 in the l th plane and its corresponding normal η l . Thenwe may have, for example, the independent constraintsη T l (x 1 − x 2 ) = 0η T l (x 2 − x 3 ) = 0(761)expressible in terms of constant symmetric matrices A ;〈Z , A l12 〉 = 0〈Z , A l23 〉 = 0(762)NMR data is noisy, so measurements are described by given upper and lowerbounds although normals η can be constrained to be exactly unit length;δ(G ℵ ) = 1 (763)