# 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.

• This bears a faint similarity to a blind image deblurring problem, but it's strange that the regularization term encourages $u \approx L(u)$. – littleO Oct 13 '17 at 0:12
• Agreed, that's a very interesting choice. I'm wondering if it perhaps it is meant to reduce overshoot. – Michael Grant Oct 13 '17 at 22:34
• Well, you're still applying an operator to $u$, namely $I-L$, so it's not really so unusual – Jason Born Aug 8 '18 at 22:12

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.