v2009.01.01 - Convex Optimization
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v2009.01.01 - Convex Optimization

v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization

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300 CHAPTER 4. SEMIDEFINITE PROGRAMMING gap-loss easy: it amounts to solving a linear system of equations and requires little or no optimization; with caveat, number of equations exceeds cardinality of signal representation (roughly l ≥ k) with respect to DCT basis. But addition of a significant amount of noise η increases level of difficulty dramatically; a 1-norm based method of reducing cardinality, for example, almost always returns DCT basis coefficients numbering in excess of minimum cardinality. We speculate that is because signal cardinality 2l becomes the predominant cardinality. DCT basis coefficient cardinality is an explicit constraint to the optimization problem we shall pose: In presence of noise, constraints equating reconstructed signal f to received signal g are not possible. We can instead formulate the dropout recovery problem as a best approximation: ∥[ ∥∥∥ minimize x∈R n subject to f = Ψx x ≽ 0 cardx ≤ k ]∥ f 1:l − g 1:l ∥∥∥ f n−l+1:n − g n−l+1:n (689) We propose solving this nonconvex problem (689) by moving the cardinality constraint to the objective as a regularization term as explained in4.5; id est, by iteration of two convex problems until convergence: and minimize 〈x , y〉 + x∈R n subject to f = Ψx x ≽ 0 [ ∥ minimize y∈R n 〈x ⋆ , y〉 subject to 0 ≼ y ≼ 1 ]∥ f 1:l − g 1:l ∥∥∥ f n−l+1:n − g n−l+1:n y T 1 = n − k (467) (690) Signal cardinality 2l is implicit to the problem statement. When number of samples in the dropout region exceeds half the window size, then that deficient cardinality of signal remaining becomes a source of degradation to reconstruction in presence of noise. Thus, by observation, we divine a reconstruction rule for this signal dropout problem to attain good

4.5. CONSTRAINING CARDINALITY 301 flatline and g 0.25 0.2 0.15 s + η 0.1 0.05 0 −0.05 −0.1 η s + η (a) −0.15 −0.2 dropout (s = 0) −0.25 0 100 200 300 400 500 f and g 0.25 0.2 0.15 f 0.1 0.05 0 (b) −0.05 −0.1 −0.15 −0.2 −0.25 0 100 200 300 400 500 Figure 81: (a) Signal dropout in signal s corrupted by noise η (SNR =10dB, g = s + η). Flatline indicates duration of signal dropout. (b) Reconstructed signal f (red) overlaid with corrupted signal g .

4.5. CONSTRAINING CARDINALITY 301<br />

flatline and g<br />

0.25<br />

0.2<br />

0.15<br />

s + η<br />

0.1<br />

0.05<br />

0<br />

−0.05<br />

−0.1<br />

η<br />

s + η<br />

(a)<br />

−0.15<br />

−0.2<br />

dropout (s = 0)<br />

−0.25<br />

0 100 200 300 400 500<br />

f and g<br />

0.25<br />

0.2<br />

0.15<br />

f<br />

0.1<br />

0.05<br />

0<br />

(b)<br />

−0.05<br />

−0.1<br />

−0.15<br />

−0.2<br />

−0.25<br />

0 100 200 300 400 500<br />

Figure 81: (a) Signal dropout in signal s corrupted by noise η (SNR =10dB,<br />

g = s + η). Flatline indicates duration of signal dropout. (b) Reconstructed<br />

signal f (red) overlaid with corrupted signal g .

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