sparse image representation via combined transforms - Convex ...

4.12. PROOFS 99
4.12.2 Proof of Theorem 4.1
We will prove convergence first, and then prove that the limiting distribution is x ′ .
When γ takes all the real positive values, (x(γ),γ) forms a continuous and differentiable
trajectory in R N+1 .Letx i γ denote the ith element of x(γ). By the first-order condition, we
have
⎛ ⎞
∂ ¯ρ(x 1 γ ,γ)
∂x
0=−2Φ T (y − Φx(γ)) + λ ⎜ ⎟
⎝ ⎠ .
.
∂ ¯ρ(x N γ ,γ)
∂x
Taking d
dγ
Since
on both sides, we have
0=2Φ T Φ dx(γ)
dγ
⎛
+ λ ⎜
⎝
∂ 2 ¯ρ(x 1 γ ,γ)
∂γ∂x
∂ 2 ¯ρ(x N γ ,γ)
∂γ∂x
+ dx1 γ
dγ
.
+ dxN γ
dγ
∂ 2 ¯ρ(x 1 γ ,γ)
∂x∂x
∂ 2 ¯ρ(x N γ ,γ)
∂x∂x
⎞
⎟
⎠ .
∂ 2 ¯ρ(x, γ)
∂γ∂x
= xe −γ|x| ,
and
∂ 2 ¯ρ(x, γ)
∂x∂x
= γe −γ|x| ,
we have
−2Φ T Φ dx(γ)
dγ
⎛
= λ ⎜
⎝
⎞
x 1 γe −γ|x1γ| + dx1 γ
γ|
dγ γe−γ|x1 .
⎟
⎠ .
x N γ e −γ|xN γ | + dxN γ
dγ γe−γ|xN γ |
Based on Assumption 2, suppose that the kth entry of vector dx(γ)
dγ
the maximum amplitude of the vector:
dx(γ)
∥ dγ ∥ = − dxk γ
∞
dγ .
is negative and takes