Recently, I was reading Efficient Algorithms for Convolutional Sparse Representations, one paper about convolutional representations.
I am interested in the optimization problem mentioned in the paper: \begin{equation} \min_{\{\boldsymbol{x}_{m}\}}~\frac{1}{2}\left\|\sum_{m}\boldsymbol{d}_{m}\star\boldsymbol{x}_{m}-\boldsymbol{s}\right\|_{2}^{2}+\frac{\rho}{2}\sum_{m}\|\boldsymbol{x}_{m}-\boldsymbol{z}_{m}\|_{2}^{2} \end{equation} where $\star$ denotes the operator of convolution.
The paper gives a solution by using the Discrete Fourier Transform (DFT) domain formulation. The paper considers to
- define linear operators $\boldsymbol{D}_{m}$ such that $\boldsymbol{D}_{m}\boldsymbol{x}_{m}=\boldsymbol{d}_{m}\star\boldsymbol{x}_{m}$,
- and introduce the variables $\{\hat{\boldsymbol{D}}_{m},\hat{\boldsymbol{x}}_{m},\hat{\boldsymbol{s}},\hat{\boldsymbol{z}}_{m}\}$ referring to the variables $\{{\boldsymbol{D}}_{m},{\boldsymbol{x}}_{m},{\boldsymbol{s}},{\boldsymbol{z}}_{m}\}$ in the DFT domain.
By doing so, it is possible to utilize the DFT convolution theorem, and we therefore have \begin{equation} \min_{\hat{\boldsymbol{x}}_{m}}~\frac{1}{2}\left\|\sum_{m}\hat{\boldsymbol{D}}_{m}\hat{\boldsymbol{x}}_{m}-\hat{\boldsymbol{s}}\right\|_{2}^{2}+\frac{\rho}{2}\|\hat{\boldsymbol{x}}_{m}-\hat{\boldsymbol{z}}_{m}\|_{2}^{2} \end{equation}
In this case, it is not hard to write down the closed-form solution to $\{\hat{\boldsymbol{x}}_{m}\}$. The paper claims that the inverse DFT of $\{\hat{\boldsymbol{x}}_{m}\}$ is the solution to $\{\boldsymbol{x}_{m}\}$.
My question is: Why should we use the DFT domain formulation? If the original optimization problem can be written as follows, \begin{equation} \min_{\{\boldsymbol{x}_{m}\}}~\frac{1}{2}\left\|\sum_{m}\boldsymbol{D}_{m}\boldsymbol{x}_{m}-\boldsymbol{s}\right\|_{2}^{2}+\frac{\rho}{2}\sum_{m}\|\boldsymbol{x}_{m}-\boldsymbol{z}_{m}\|_{2}^{2} \end{equation} then it should be not hard to obtain the closed-form solution to $\{\boldsymbol{x}_{m}\}$ from this optimization problem.
My question may be naive, but it confused me a lot. Thank you for your help in advance!