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Reconstruction by Convex Optimization under Low Rank and Cardinality

Reconstruction by Convex Optimization under Low Rank and Cardinality. Jon Dattorro. prototypical cardinality problem. Combinatorial Geometric. Perspectives:. Euclidean bodies. Permutation Polyhedron. n ! permutation matrices are vertices in ( n -1) 2 dimensions.

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Reconstruction by Convex Optimization under Low Rank and Cardinality

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  1. Reconstruction by Convex Optimization under Low Rank and Cardinality Jon Dattorro convexoptimization.com

  2. prototypical cardinality problem • Combinatorial • Geometric Perspectives:

  3. Euclidean bodies Permutation Polyhedron • n! permutation matrices are vertices in (n-1)2dimensions. • permutaton matrices are minimum cardinality doubly stochastic matrices. Hyperplane

  4. Geometrical perspective Compressed Sensing 1-norm ball: 2n vertices, 2n facets Candes/Donoho (2004)

  5. Candes demo • %Emmanuel Candes, California Institute of Technology, June 6 2007, IMA Summerschool. • clear all, close all • n = 512; % Size of signal • m = 64; % Number of samples (undersample by a factor 8) • k = 0:n-1; t = 0:n-1; • F = exp(-i*2*pi*k'*t/n)/sqrt(n); % Fourier matrix • freq = randsample(n,m); • A = [real(F(freq,:)); • imag(F(freq,:))]; % Incomplete Fourier matrix • S = 28; • support = randsample(n,S); • x0 = zeros(n,1); x0(support) = randn(S,1); • b = A*x0; • % Solve l1 using CVX • cvx_quiet(true); • cvx_begin • variable x(n); • minimize(norm(x,1)); • A*x == b; • cvx_end • norm(x - x0)/norm(x0) • figure, plot(1:n,x0,'b*',1:n,x,'ro'), legend('original','decoded') wikimization.org

  6. Candes demo

  7. k-sparse sampling theorem • Donoho/Tanner (2005)

  8. two geometrical interpretations

  9. motivation to study cones • convex cones generalize orthogonal subspaces • Projection on K determinable from projection on -K* and vice versa. (Moreau) • Dual cone:

  10. application - LP presolver • Delete rows and columns of matrix A • columns: smallest face F of cone K containing b • A holds generators for K • If feasible, throw A(: , i) away

  11. application - Cartography

  12. list reconstruction from distance D a.k.a • metric multidimensional scaling • principal component analysis • Karhunen-Loeve transform • cartography: projection on semidefinite cone

  13. projection on semidefinite cone because subspace of symmetric matrices is isomorphic with subspace of symmetric hollow matrices

  14. is convex problem (Eckart & Young) (§7.1.4 CO&EDG) • optimal list X from (§5.12 CO&EDG) (EY)

  15. ordinal reconstruction • nonconvex • strategy: break into two problems: (EY) and convex problem • fast projection on monotone nonnegative cone KM+(Nemeth, 2009)

  16. Cardinality heuristics

  17. Rank heuristics • trace is convex envelope of rank on PSD matrices • rank function is quasiconcave

  18. Idea behind convex iteration (vector inner product)

  19. Convex Iteration

  20. application - (Recht, Fazel, Parrilo, 2007) (Rice University 2005)

  21. one-pixel camera - MIT

  22. one-pixel camera - MIT

  23. application - MRI phantom • Led directly to sparse sampling theorem MATLAB>> phantom(256) Candes, Romberg, Tao 2004

  24. application - MRI phantom • MRI raw data called k-space • Raw data in Fourier domain • aliasing at 4% subsampling

  25. application - MRI phantom (projection matrix) • hard to compute y is direction vector from convex iteration

  26. application - MRI phantom

  27. application - MRI phantom reconstruction error: -103dB

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