Multitask multivariate regression? 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 $L &gt; M &gt;&gt; N$

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?
You may want to model your problem using bayesian regression, which may allow you to introduce your prior knowledge in the form of priors (a priori distributions) of the model parameters. They would also allow you to model latent variables that govern the dynamics of the interactions (and impose priors on them as well).

The specific approach may be based on sampling (e.g. Markov Chain Monte Carlo) or optimization (e.g. Variational Bayes).

One of the most popular bayesian frameworks is Stan, which has bindings to R (rstan) and python (pystan). In R there are other alternatives such a BUGS and JAGS. In the python realm other options are PyMC (which is also pretty popular) or Edward.