Weighting operators for sparsity regularization

Case 1: 𝐲 = c ⁢ 𝐱 α , c ≠ 1 .

By the convexity of the cost-functional in (3.7) and the argument presented above, it follows that 𝐲 = c ⁢ 𝐱 α cannot be a minimizer unless c = 1 .

Case 2: 𝐲 ≠ c ⁢ 𝐱 α .

In this case there must exist at least one component y k , k ≠ j , of 𝐲 such that y k ≠ 0 . Consider

Also, by the definition of the subdifferential,

for any 𝐳 ∈ ∂ ⁡ h ⁢ ( 𝖶 ⁢ 𝐱 α ) . Consequently, we get

Recall that 𝐱 α = γ α ⁢ 𝐞 j . From Lemma (3.2) , we can write (A.2) as

(A.5)

This implies that

However, choosing 𝐳 = 1 α ⁢ 𝖶 - 1 ⁢ 𝖢 T ⁢ 𝖢 ⁢ ( 𝐞 j - 𝐱 α ) does not immediately lead to a strict inequality in (A.4). Consequently, we must find a better choice of 𝐳 . Without loss of generality,^[1] we can assume that [ 𝖶 ⁢ 𝐲 - 𝖶 ⁢ 𝐱 α ] k > 0 and choose 𝐳 ~ = [ z ~ 1 , z ~ 2 , … , z ~ n ] T , where z ~ i is defined as

Since the condition (A.6) holds, it follows that 𝐳 ~ ∈ ∂ ⁡ h ⁢ ( 𝖶 ⁢ 𝐱 α ) .

From (A.5) we have [ 1 α ⁢ 𝖶 - 1 ⁢ 𝖢 T ⁢ 𝖢 ⁢ ( 𝐞 j - 𝐱 α ) ] k < 1 and therefore we get the strictly inequality

Finally, combining this inequality with (A.4), we obtain

where the final inequality follows from the first-order optimality conditions of the convex functional

i.e.,

This shows that 𝐱 α is the unique minimizer of 𝔍 ⁢ ( 𝐱 ) .