v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 371phantom(256)Figure 107: Shepp-Logan phantom from Matlab image processing toolbox.method (4.4.1.1), success relies on existence of matrices in the feasible setof (833) having desired rank and diagonal cardinality. In particular, thefeasible set of convex problem (833) is a Fantope (91) whose extreme pointsconstitute the set of all normalized rank-one matrices; among those are foundrank-one matrices of any desired diagonal cardinality.Convex problem (833) is neither a relaxation of cardinality problem (829);instead, problem (833) is a convex equivalent to (829) at convergence ofiteration (833) (834) (835). Because the feasible set of convex problem (833)contains all normalized (B.1) symmetric rank-one matrices of every nonzerodiagonal cardinality, a constraint too low or high in cardinality c willnot prevent solution. An optimal rank-one solution X ⋆ , whose diagonalcardinality is equal to cardinality of a principal eigenvector of matrix A ,will produce the lowest residual Frobenius norm (to within machine noiseprocesses) in the original problem statement (828).4.6.0.0.12 Example. Compressive sampling of a phantom.In summer of 2004, Candès, Romberg, & Tao [70] and Donoho [118]released papers on perfect signal reconstruction from samples that stand inviolation of Shannon’s classical sampling theorem. These defiant signals areassumed sparse inherently or under some sparsifying affine transformation.Essentially, they proposed sparse sampling theorems asserting averagesample rate independent of signal bandwidth and less than Shannon’s rate.