v2010.10.26 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 345We propose solving this nonconvex problem (773) by moving the cardinalityconstraint to the objective as a regularization term as explained in4.5;id est, by iteration of two convex problems until convergence:andminimize 〈x , y〉 +x∈R nsubject to f = Ψxx ≽ 0[∥minimizey∈R n 〈x ⋆ , y〉subject to 0 ≼ y ≼ 1]∥f 1:l − g 1:l ∥∥∥f n−l+1:n − g n−l+1:ny T 1 = n − k(525)(774)Signal cardinality 2l is implicit to the problem statement. When numberof samples in the dropout region exceeds half the window size, then thatdeficient cardinality of signal remaining becomes a source of degradationto reconstruction in presence of noise. Thus, by observation, we divinea reconstruction rule for this signal dropout problem to attain goodnoise suppression: l must exceed a maximum of cardinality bounds;2l ≥ max{2k , n/2}.Figure 101 and Figure 102 show one realization of this dropout problem.Original signal s is created by adding four (k = 4) randomly selected DCTbasis vectors, from Ψ (n = 500 in this example), whose amplitudes arerandomly selected from a uniform distribution above the noise floor; in theinterval [10 −10/20 , 1]. Then a 240-sample dropout is realized (l = 130) andGaussian noise η added to make corrupted signal g (from which a bestapproximation f will be made) having 10dB signal to noise ratio (772).The time gap contains much noise, as apparent from Figure 101a. But inonly a few iterations (774) (525), original signal s is recovered with relativeerror power 36dB down; illustrated in Figure 102. Correct cardinality isalso recovered (cardx = cardz) along with the basis vector indices used tomake original signal s . Approximation error is due to DCT basis coefficientestimate error. When this experiment is repeated 1000 times on noisy signalsaveraging 10dB SNR, the correct cardinality and indices are recovered 99%of the time with average relative error power 30dB down. Without noise, weget perfect reconstruction in one iteration. [Matlab code: Wıκımization]