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I am reading a paper Recovery of Future Data via Convolution Nuclear Norm Minimization. Here, I know there is a definition for convolution matrix.

Given any vector $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)^\top\in\mathbb{R}^{n}$ with a certain kernel size $\tau\in\mathbb{N}^+$ ($1<\tau<n$), then its convolution matrix is given by \begin{equation} \mathcal{C}_{\tau}(\boldsymbol{x})=\begin{bmatrix} x_1 & x_n & x_{n-1} & \cdots & x_{n-\tau+2} \\ x_2 & x_1 & x_n & \cdots & x_{n-\tau+3} \\ x_3 & x_2 & x_1 & \cdots & x_{n-\tau+4} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_n & x_{n-1} & x_{n-2} & \cdots & x_{n-\tau+1} \\ \end{bmatrix}\in\mathbb{R}^{n\times\tau}, \end{equation} which is also a circular matrix, but it is not a circulant matrix of size $n\times n$. In the case of circulant matrix, we can utilize fast Fourier transform to solve the nuclear norm minimization problem.

In fact, I am interested in the nuclear norm minimization of the convolution matrix, just like \begin{equation} \min_{\boldsymbol{x}}~\|\mathcal{C}_{\tau}(\boldsymbol{x})\|_{*}+\frac{\lambda}{2}\|\boldsymbol{x}-\boldsymbol{z}\|_{2}^{2} \end{equation} where $\|\mathcal{C}_{\tau}(\boldsymbol{x})\|_{*}$ is the nuclear norm of $\mathcal{C}_{\tau}(\boldsymbol{x})$, i.e., the sum of singular values. $\boldsymbol{z}$ is a known variable.

[Q] Is it possible to utilize the fast Fourier transform to solve the nuclear norm minimization of the convolution matrix?

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