sparse image representation via combined transforms - Convex ...

100 CHAPTER 4. COMBINED IMAGE REPRESENTATION
Further supposing that (Φ T Φ) kk > 0,k =1, 2,...,wehave
x k γe −γ|xk γ| + dxk γ
dγ γe−γ|xk γ|
( )
N∑
= 2(Φ T Φ) kj − dxj γ
dγ
j=1
( )
≥ 2(Φ T Φ) kk · − dxk γ
− ∑ ( )
∣
∣2(Φ T ∣
Φ) kj · − dxk γ
dγ
dγ
j≠k
( )
ɛ
≥
1+ɛ 2(ΦT Φ) kk · − dxk γ
.
dγ
Hence
dx(γ)
∥ dγ ∥ = − dxk γ
∞
dγ
≤
≤
≤
x k γe −γ|xk γ |
ɛ
1+ɛ 2(ΦT Φ) kk + γe −γ|xk γ|
1
2 √ x k γ
2√
ɛ
√ √ e −γ|xk γ|/2
1+ɛ (Φ T Φ) kk γ
1
2 √ 2
2√
ɛ
√
1+ɛ (Φ T Φ) kk
γ 3/2 e−1 .
The integration of the last term in the right-hand side of the above inequality is finite, so
the integration of dx(γ)/dγ is upper bounded by a finite quantity. Hence x(γ) converges.
When dx k γ/dγ is positive, the discussion is similar. This confirms convergence.
Now we prove the limiting vector lim γ→+∞ x(γ) isx ′ . Let x(∞) = lim γ→+∞ x(γ). Let
f(x, γ) denote the objective function in (4.9). If x ′ ≠ x(∞) and by Assumption 1 the
solution to problem (4.9) is unique, we have
there exists a fixed ε>0, f(x ′ , ∞)+ε