# Talk:Chromosome structure via Euclidean Distance Matrices

### From Wikimization

(→E.coli realization) |
(→E.coli realization) |
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Then conversion to a Euclidean distance matrix (EDM) is straightforward - <br>Chapter 5.4.2 of [http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Convex Optimization & Euclidean Distance Geometry]. | Then conversion to a Euclidean distance matrix (EDM) is straightforward - <br>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 | + | The program calculates only the first 20 eigenvalues of an oblique projection of the EDM on a positive semidefinite (PSD) cone - <br> |

+ | 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. | 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 project the EDM on | + | To get a picture corresponding to physical reality, we obliquely project the EDM on a 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 | + | It is unlikely that this picture is an accurate representation unless the number of eigenvalues of that projection approaches 3 to begin with. |

Matlab Figures allow 3D rotation in real time, so you can get a good idea of the body's shape. | 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. | I include a low-resolution figure here (frame 4) for reference. |

## Revision as of 20:55, 7 August 2008

%%% Ronan Fleming, E.coli molecule data %%% -Jon Dattorro, August 2008 clear all load ecoli frame = 4; % 1 through 12 G = her49imfs12movfull(frame).cdata; % uint8 G = (double(G)-128)/128; % Gram matrix N = size(G,1); D = diag(G)*ones(N,1)' + ones(N,1)*diag(G)' - 2*G; % EDM D clear her49imfs12movfull G; Vn = [-ones(1,N-1); speye(N-1)]; VDV = (-Vn'*D*Vn)/2; clear D Vn [evec evals flag] = eigs(VDV, [], 20, 'LA'); if flag, disp('convergence problem'), return, end; close all Xs = 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

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 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 *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 a 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 to begin with.

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.