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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?

I would appreciate any reference or insight about this kind of problem.

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  • $\begingroup$ Mathematically, this question makes no sense, because you literally are asking to factor a matrix of distributions as a product of matrices of random variables. There is an inherent difficulty with this, in that there are many different possible interpretations of the algebraic operations of multiplication and addition involved in this product. Could you provide some context that might help us guess what you're trying to ask? If not, then it is crucial that your mathematical formulation be perfectly correct and unambiguous. $\endgroup$ – whuber May 17 '19 at 15:23