v2009.01.01 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 303
noise suppression: l must exceed a maximum of cardinality bounds;
2l ≥ max{2k , n/2}.
Figure 81 and Figure 82 show one realization of this dropout problem.
Original signal s is created by adding four (k = 4) randomly selected DCT
basis vectors, from Ψ (n = 500 in this example), whose amplitudes are
randomly selected from a uniform distribution above the noise floor; in the
interval [10 −10/20 , 1]. Then a 240-sample dropout is realized (l = 130) and
Gaussian noise η added to make corrupted signal g (from which a best
approximation f will be made) having 10dB signal to noise ratio (688).
The time gap contains much noise, as apparent from Figure 81a. But in
only a few iterations (690) (467), original signal s is recovered with relative
error power 36dB down; illustrated in Figure 82. Correct cardinality is
also recovered (cardx = cardz) along with the basis vector indices used to
make original signal s . Approximation error is due to DCT basis coefficient
estimate error. When this experiment is repeated 1000 times on noisy signals
averaging 10dB SNR, the correct cardinality and indices are recovered 99%
of the time with average relative error power 30dB down. Without noise, we
get perfect reconstruction in one iteration. (Matlab code: Wıκımization)
4.5.1.3 Compressed sensing geometry with a nonnegative variable
It is well known that cardinality problem (682), on page 294, is easier to solve
by linear programming when variable x is nonnegatively constrained than
when not; e.g., Figure 62, Figure 80. We postulate a simple geometrical
explanation: Figure 61 illustrates 1-norm ball B 1 in R 3 and affine subset
A = {x |Ax=b}. Prototypical compressed sensing problem, for A∈ R m×n
minimize ‖x‖ 1
x
subject to Ax = b
(460)
is solved when the 1-norm ball B 1 kisses affine subset A .
If variable x is constrained to the nonnegative orthant,
minimize ‖x‖ 1
x
subject to Ax = b
x ≽ 0
then 1-norm ball B 1 (Figure 61) becomes simplex S in Figure 83.
(691)