v2010.10.26 - Convex Optimization

388 CHAPTER 4. SEMIDEFINITE PROGRAMMING1 2 3 4 5samples0 1000 2000 3000 3726Figure 114: Estimates of compression for various encoding methods:1) linear interpolation (140 samples),2) minimal columnar basis (311 samples),3) convex iteration (700 samples) can achieve lower bound predicted bycompressed sensing (670 samples, n=46×81, k =140, Figure 100) whereasnuclear norm minimization alone does not [300,6],4) JPEG @100% quality (2588 samples),5) no compression (3726 samples).programwith semidefinite programminimizeW 1 ∈ S 46 , W 2 ∈ S 81 , X ∈ R 46×81subject tominimizeZ ∈ S 46+81([ ] )W1 XtrX T ZW 2A vec X = y[ ]W1 XX T ≽ 0W 2([ ] ⋆ )W1 XtrX T ZW 2subject to 0 ≼ Z ≼ ItrZ = 46 + 81 − 5(867)(868)(which has an optimal solution that is known in closed form, p.658) until arank-5 composite matrix is found.With 1000 samples {y i } , convergence occurs in two iterations; 700samples require more than ten iterations but reconstruction remainsperfect. Iterating more admits taking of fewer samples. Reconstruction isindependent of pseudorandom sequence parameters; e.g., binary sequencessucceed with the same efficiency as Gaussian or uniformly distributedsequences.