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

2. prototypical cardinality problem
Perspectives:
Combinatorial
Geometric

3. Euclidean bodies Permutation Polyhedron
n! permutation matrices are vertices in (n-1)2 dimensions.
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 wikimization.org
%Emmanuel Candes, California Institute of Technology, June , 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)
(EY)
is convex problem (Eckart & Young) (§7.1.4 CO&EDG)
optimal list X from (§5.12 CO&EDG)

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