The name is absent

that determine λ. When there is no noise in measurements, i.e., Ax_r = b, the
regularization coefficient λ should be set infinity. As measurement noise increases,
we should relax t₂^^norm constraint or equivalently decrease λ. In addition, sparse
vectors imply small λ. When it is known that vector x is strongly sparse, one
should relax t⅛-norm constraint (decrease λ) to obtain a very sparse solution for
the problem.

Figure 4.5 shows estimation error for different regularization coefficients, λ.
As explained, for very small λ and very large λ estimation error is high. There is
an optimal regularization coefficient λ_opt in which the variation estimation error is
minimum. Optimizing Equation 4.13 for λ = λ_opt leads to the minimum variation
estimation error, λ_opt is a function of the measurement matrix, measurement
noise, and the true variations x_r∙, thus, it is not possible to find λ_opt exactly.

Applying first-order necessity condition for regularization problem in Equa-
tion 4.13 determines minimum value for λ. Let

j(æ)=   ∣∣τ∣∣ι + λ∣∣Ar - b∣∣∣.

The first-order necessity condition for optimal solution implies = 0, i =
1... n. Thus,

g∣∣x∣∣₁ _ ∂λ∣∣Ar-⅛∣∣j

∂Xi           ∂Xi

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