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I want to make a prediction model for the following transformation

$f:\mathbb{N}^N \rightarrow \mathbb{R}^M$

$N$ is around 20 and vectors $\mathbf{v} \in \mathbb{N}^N$ are sparse with $\sum_{i=1}^N v_i = 3$. I suspect that the transformation is stateless and each component $v_i$ triggers random generators for specific components $u_j$ of $\mathbf{u} \in \mathbb{R}^M$, with $M$ around 6. I have a fixed set of observations. What is the best way to predict the influence of each component $v_i$?

What I tried was to concatenate input and output vectors and calculate the covariance matrix for the resulting vectors.

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