# Talk:Chromosome structure via Euclidean Distance Matrices

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-                                                          %%% Ronan Fleming, E.coli molecule data
-                                                          %%% -Jon Dattorro, August 9 2008
-                                                          clear all

-                                                          frame = 4;                                          % 1 through 12
-                                                          G = her49imfs12movfull(frame).cdata;                % uint8
-                                                          G = (double(G)-128)/128;                            % Gram matrix
-                                                          N = size(G,1);
-
-                                                          Vn = [-ones(1,N-1); speye(N-1)];
-                                                          [evec evals flag] = eigs(Vn'*G*Vn, [], 20, 'LA');
-                                                          if flag, disp('convergence problem'), return, end;
-
-                                                          close all
-                                                          Xs = [zeros(3,1) sqrt(real(evals(1:3,1:3)))*real(evec(:,1:3))'];  % Projection of -VDV on PSD cone rank 3
-                                                          plot3(Xs(1,:), Xs(2,:), Xs(3,:), '.')
-
- - == E.coli realization == - [[Image:E.coli-4.jpg|thumb|right|560px|Test image E.coli]] - I regard the autocorrelation data you provided as a Gram matrix. - - Then conversion to a Euclidean distance matrix (EDM) is straightforward -
Chapter 5.4.2 of [http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Convex Optimization & Euclidean Distance Geometry]. - - The program calculates only the first 20 eigenvalues of an oblique projection of the EDM on a positive semidefinite (PSD) cone -
- Chapter 7.0.4 - 7.1 [http://meboo.convexoptimization.com/BOOK/ProximityProblems.pdf ''ibidem'']. - - You can see at runtime that there are many significant eigenvalues; which means, the Euclidean body (the molecule) lives in a space higher than dimension 3, assuming I have interpreted the E.coli data correctly. - - To get a picture corresponding to physical reality, we obliquely project the EDM on the closest rank-3 subset of the boundary of that PSD cone; this means, precisely, we truncate eigenvalues. - - It is unlikely that this picture is an accurate representation unless the number of eigenvalues of that projection approaches 3 prior to truncation. - - Matlab Figures allow 3D rotation in real time, so you can get a good idea of the body's shape. - - I include a low-resolution figure here (frame 4) for reference.