v2009.01.01 - Convex Optimization

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