v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 381this is equivalent to(lim Ψ T δ(y)δ(|Ψ vec U| + ǫ1) −1 Ψ + λP ) vec U = λPf (859)ǫ→0where small positive constant ǫ ∈ R + has been introduced for invertibility.The analytical effect of introducing ǫ is to make the objective’s gradient valideverywhere in the function domain; id est, it transforms absolute value in(856b) from a function differentiable almost everywhere into a differentiablefunction. An example of such a transformation in one dimension is illustratedin Figure 111. When small enough for practical purposes 4.56 (ǫ≈1E-3), wemay ignore the limiting operation. Then the mapping, for 0 ≼ y ≼ 1vec U t+1 = ( Ψ T δ(y)δ(|Ψ vec U t | + ǫ1) −1 Ψ + λP ) −1λPf (860)is a contraction in U t that can be solved recursively in t for itsunique fixed point; id est, until U t+1 → U t . [227, p.300] [203, p.155]Calculating this inversion directly is not possible for large matrices oncontemporary computers because of numerical precision, so instead we applythe conjugate gradient method of solution to(Ψ T δ(y)δ(|Ψ vec U t | + ǫ1) −1 Ψ + λP ) vec U t+1 = λPf (861)which is linear in U t+1 at each recursion in the Matlab program. 4.57Observe that P (848), in the equality constraint from problem (856a), isnot a fat matrix. 4.58 Although number of Fourier samples taken is equal tothe number of nonzero entries in binary mask Φ , matrix P is square butnever actually formed during computation. Rather, a two-dimensional fastFourier transform of U is computed followed by masking with ΘΦΘ andthen an inverse fast Fourier transform. This technique significantly reducesmemory requirements and, together with contraction method of solution, isthe principal reason for relatively fast computation.4.56 We are looking for at least 50dB image/error ratio from only 4.1% subsampled data(10 radial lines in k-space). With this setting of ǫ, we actually attain in excess of 100dBfrom a simple Matlab program in about a minute on a 2006 vintage laptop Core 2 CPU(Intel T7400@2.16GHz, 666MHz FSB). By trading execution time and treating discreteimage-gradient cardinality as a known quantity for this phantom, over 160dB is achievable.4.57 Conjugate gradient method requires positive definiteness. [152,4.8.3.2]4.58 Fat is typical of compressed sensing problems; e.g., [69] [76].