sparse image representation via combined transforms - Convex ...

98 CHAPTER 4. COMBINED IMAGE REPRESENTATION
and its corresponding version with a Lagrangian multiplier λ,
(RIA λ )
x (k+1) = argmin ‖y − Φx‖ 2 2 + λ
x
N∑
i=1
|x i |
|x (k)
i
| + δ .
We know the following results.
Theorem 4.2 Suppose we add one more constraint on x: x i ≥ 0,i=1, 2,... ,N. The sequence
generated by (RIA), {x (k)
i
,k =1, 2, 3,...}, converges in the sense that the difference
of sequential elements goes to zero:
|x (k+1)
i
− x (k)
i
|→0, as k → +∞.
We learned this result from [95].
Theorem 4.3 If the sequence {x (k)
i
,k = 1, 2, 3,...} generated by (RIA λ ) converges, it
converges to a local minimum of (LO λ ).
Some related works can be found in [36, 106]. There is also some ongoing research, for
example, the work being carried out by Boyd, Lobo and Fazel in the Information Systems
Laboratories, Stanford.
4.12 Proofs
4.12.1 Proof of Proposition 4.1
When ̂x is the solution to the problem as in (4.2), the first order condition is
0=λ∇ρ(x)+2Φ T Φx − 2Φ T y,
where ∇ρ(x) is the gradient vector of ρ(x). Hence
1
2 λ∇ρ(x) =ΦT (y − Φx) . (4.15)
Recall ρ(x) = ∑ N
i=1 ¯ρ(x i). It’s easy to verify that ∀x i , |¯ρ ′ (x i )|≤1. Hence for the left-hand
side of (4.15), we have ‖ 1 2 λ∇ρ(x)‖ ∞ ≤ λ 2
. The bound (4.6) follows. ✷