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I have two sets of vectors, A and B. Vectors from set A live in an m-dimensional space, while those from set B are n-dimensional.

I also have a mapping f from A to B. I would like to learn an n x m matrix Q such that the following sum is minimized:

$$ \sum_{v \in A}{\|Qv - f(v)\|^2}$$

Are there any standard approaches to tackle this, for example in Matlab?

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Just do multiple linear regression for each of the $n$ dimensions independently and take each set of $m$ coefficients as a row in $Q$. Since square root is monotone for positive real numbers it is not hard to convince yourself that if for all $i$, $Q_{i,*}\in \mathbb{R}^m$ minimizes

$$ \sum_{v \in A} \left\|Q_{i,*}v - f(v)_i\right\|^2 $$

Then $Q \in \mathbb{R}^{n \times m}$, where each $Q_{i,*}$ is a row in $Q$, minimizes

$$ \sum_{v \in A} \left\|Qv - f(v)\right\|^2. $$

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