I'm trying to solve a multivariate regression problem similar to PLS regression.

The problem can be described as a connectivity analysis problem where we have two regions with unknown unidirectional connections(many-to-many) and given a set of input region patterns and output region patterns, we want to infer the underlying connections.

Mathematically, the problem can be formulated as below

$Y = BX \qquad$ where $Y \in \mathbb{R}_+^{M\times N}$, $X \in \mathbb{R}_+^{L\times N}$, and $B \in \mathbb{R}_+^{M\times L}$ with $L > M >> N$

The column of $X$ and $Y$, will be a vectorized version of 2D image.

Although this would result in highly underdetermined system, I do have some prior knowledge about the pattern in input/output regions that I can incorporate in the model.

Is there a model/idea that I can use in situation like this?