Factorizing a matrix of distributions [closed]

Let's say we have a matrix $$X \in \mathbb{R}^{m \times n}$$, then the (R-truncated) SVD allows to approximate:

$$X_{i,j} \approx \sum\limits_{r=1}^{R} \sigma_r \times U_{r,i} \times V_{r, j}$$

Now I want to consider the case where $$X_{i,j}$$ is not a scalar, but a distribution. Let's say $$X_{i,j} = \mathcal{N}(\mu_{i,j}, \sigma_{i,j})$$ (note that I do not mean to say that $$X_{i,j} \sim \mathcal{N}(\mu_{i,j}, \sigma_{i,j})$$, but that $$X_{i,j}$$ $$\textit{is}$$ that distribution).

Is there any model that could help me factor such a matrix of distributions? For example, would it be possible to approximate:

$$X_{i,j} \approx \sum\limits_{r=1}^{R} \sigma_r \times U_{r,i} \times V_{r, j}$$

where in this case $$\approx$$ could be a measure of distance between distributions (e.g. KL divergence) and $$U_{r,i}$$ and $$V_{r,j}$$ could be random variables instead of constants?