Consider the cost function
$$f(X,\lambda) = \|AX-b\|_2^2 + \alpha \|LX\|_2^2$$
$A:$Measurement matrix($R_{m\times n}$,$m \ll n$), $b:$observation vector($R_m$), $L:$Laplacian operator($R_{n \times n}$), $X:$vector form of an image($R_n$)
(The above cost function can be read, non-mathematically as minimize
$AX=b+error$ subject to $LX=0$)
$$\frac{\partial f}{dX} = 0 \implies 2X^T(A^TA) - 2X^T(A^Tb)+\lambda(L^TL) = 0.........(1)$$
$$\frac{\partial f}{d\lambda} = 0 \implies \|LX\|_2^2=0.........(2)$$
From $(1)$ $X_s = (A^TA+\lambda L^TL)^{-1}A^Tb.........(3)$
$(3)$ in $(2) \implies \|LX_s\|^2_2 = 0..........(4)$
It is very clear from $(3),$ $\lambda$ plays a very crucial role in solving for $X$ because changing the value of $\lambda$ gives a different estimate for $X$.
using $(4),$ is it not possible to auto compute the optimal value of $\lambda$?
If it is possible to auto compute optimal $\lambda,$ why is it in many image de-noising problems we specify the optimal lambda $\lambda$ as an input to the algorithm?
It it is not possible to auto compute $\lambda,$ why is it so?
Thanks for reading my question with patience..