v2007.09.17 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 289Our approach to reconstruction is to look for low-rank solution to anunderdetermined system:find Xsubject to A vec X = yrankX ≤ 5(693)where vec X is vectorized matrix X ∈ R 46×81 (stacked columns). Each rowof fat matrix A is one realization of a pseudorandom pattern applied to themicromirrors. Since these patterns are deterministic (known), then the i thsample y i equals A(i, :) vec Y ; id est, y = A vec Y . Perfect reconstructionmeans optimal solution X ⋆ equals scene Y ∈ R 46×81 to within machineprecision.Because variable matrix X is generally not square or positive semidefinite,we constrain its rank by rewriting the problem equivalentlyfind Xsubject to A vec[X = y]W1 XrankX T ≤ 5W 2(694)This rank constraint on the composite matrix insures rankX ≤ 5 for anychoice of dimensionally compatible matrices W 1 and W 2 . But to solve thisproblem by convex iteration, we alternate solution of semidefinite programwith semidefinite programminimizeW 1 , W 2 , X[ ]W1 XX T ZW 2subject to A vec X = y[ ]W1 XX T ≽ 0W 2(695)minimizeZ[ ] ⋆W1 XX T ZW 2subject to 0 ≼ Z ≼ ItrZ = 46 + 81 − 5(696)(whose solution has closed form, p.541) until a rank-5 composite matrixis found. With 1000 samples {y i } , convergence occurs in two iterations;