optimize weight coefficient: $\text{arg} \min_{\beta}\|U- \sum_{i=1}^{k} \beta_i V_i \|_F + \lambda\|\beta\|_2^2$

Question

I would like to optimize the follow objective function, where $U \in \mathbb{R}^{m \times n}$, $V_i \in \mathbb{R}^{m \times n}$, $\beta \in \mathbb{R}^{k}$.

Actually, $\beta$ is the weight of each $V$, what is the algorithm to get the optimal $\beta$? Here the first term is Frobenius norm and second term is $2$-norm, I am not familiar with this kinds of objective function, can someone help me understand this objective function? what existing software (R or Matlab) can solve this? or need code by myself? Is this similar to ridge regression or lasso or others?

Thanks.

$$\text{arg} \min_{\beta}\left\|U- \sum_{i=1}^{k} \beta_i V_i \right\|_F + \lambda\|\beta\|_2^2$$

Answer 1

First, change $U$ to $R^{mn \times 1}$, called $y$, and for each of $V_1,...,V_k$ also change to $R^{mn \times 1}$ and column bind to form a new matrix, say $X$.

then the queation is ridge regression problem and the solution for $beta$ is:

$$(X^TX + \lambda I)^{-1} X^T y $$

thanks @Arin Chaudhuri.