SVD with Laplacian regularization and $L_{1,2}$ group-norm

Question

I have a data matrix of the form $X \in \mathbb{R}^{n\times m}$ where the $n$ rows have spatial relationships and $m$ columns have temporal relationships. I am trying to model an objective function of the form

$|| X - U\Sigma V^T||_F^2 + ||V^TR||_{1,2} + \mathbf{tr}(U^TLU) + ||U||_1 + ||V||_1$

s.t $U,V > 0$ and $\Sigma = diag([\sigma_1,\sigma_2,..,\sigma_r]) $

Here,

• $\Sigma \in \mathbb{R}^{r\times r}$ is a diagonal matrix.
• $L \in \mathbb{R}^{n\times n}$ is a graph Laplacian.
• $U \in \mathbb{R}^{n\times r}$ and $V \in \mathbb{R}^{m\times r}$.
• $R \in \{-1,1,0\}^{m \times m-1}$

The $R$ matrix has -1's on its primary diagonal. 1 on its second diagonal and zeros everywhere else. Similar to

taken from the paper titled Subspace Clustering for Sequential Data

I have two questions:

1. Can the Frobenius norm term above i.e. $||X - U\Sigma V||_F^2$ be considered a singular value decomposition of $X$?

2. It would be helpful to understand the approach to solving the above objective function and any guidance with links to a blog or research article that may help me address this problem would be much appreciated.

Thanks in advance!