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Reconstruction by Convex Optimization under Low Rank and Cardinality

This document delves into the intricate relationship between convex optimization and problems characterized by low rank and cardinality. It examines the prototypical aspects of cardinality problems from combinatorial and geometric perspectives, focusing on permutation matrices and their representations in Euclidean spaces. The framework includes compressed sensing techniques and recovery methodologies utilizing l1-norm minimization. Notable applications in MRI and other imaging modalities are discussed, further illustrating the advantages of applying these convex optimization strategies to enhance signal recovery and data reconstruction.

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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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