What type of minimization problem is $\arg \min_u \{\|s-u\|+\lambda\|u-L\left(u\right)\|\}$? A few years ago, I came across the following minimization formulation
$$J = \arg \min_u \{\|s-u\|+\lambda\|u-L\left(u\right)\|\}$$
$s$ is the measurement, $\lambda$ is the regularization parameter, and $L\left(u\right)$ is a (local) convolution with adaptive coefficients in its kernel, which will be found iteratively. After these coefficients are updated, $u$ will also be updated by minimizing $J$. These steps will be looped until convergency is reached in both the coefficients of $L$ and the solution $u$.
Unfortunately, I can't find the right keywords to search for the method. Could someone help me find a paper, webpage, the method's name, or any hint so that I can read and study the method in more detail? Thank you.
 A: Consider
\begin{align}
Ax = u
\end{align}
Because if
\begin{align}
L = \frac{Axx^TA^T}{x^TA^TAx}
\end{align}
We see that
\begin{align}
\left\lVert \left(I-L\right)u \right\rVert &= \left\lVert \left(I-\frac{Axx^TA^T}{x^TA^TAx}\right)u \right\rVert\\
&=\left\lVert\left(I-\frac{uu^T}{\left\lVert u\right\rVert^2}\right)u \right\rVert\\
&=0
\end{align}
Here is where it would be useful:
We wish to describe $u$ by a weighted superposition of model matrices $A_i$ via parameters $x$, i.e.
\begin{align}
u &= B_ax\\
&= \left(\sum a_i A_i\right) x
\end{align}
Then optimizing over both $x$ and $a_i$ means that you are asking the question:
"Given the data, what parameters $x$ do I have to estimate, and what model $B_a$ do I have to choose to describe the data the best"
If $s$ can be described by $B_ax$ completely, then we will also have $s=u$ in the end. I guess the regularization parameter doesn't really serve a purpose in this case. Correct me if I'm wrong.
But I guess this kind of objective function would be used in circumstances where the operators $a_i$ coefficients are not perfectly known. For example in inverse problems this would be useful, because approximate (wrong) models can yield formation of artifacts in the parameters $x$, which you want to correctly estimate.
I do not know the right keywords either to look for it, but would also appreciate any further comments hinting in this direction.
