This may be odd, but say someone solved $d = Gm$ using the least squares method and calculated the parameters vector $m$. I have the vector of data $d$, how could I solve for $G$ to see the matrix that was used with something like Python?


Nothing stops you from solving the Least Squares problem:

$$ \arg \min_{A} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} $$

If we set $ f \left( A \right) = \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} $ then:

$$ \frac{\mathrm{d} f \left( A \right) }{\mathrm{d} A} = \left( A x - b \right) {x}^{T} $$

Solving for the case will give you:

$$ \arg \min_{A} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} = {\left( x {x}^{T} \right)}^{-1} b {x}^{T} $$

But the issues is that $ \left( x {x}^{T} \right) $ is a matrix of rank 1. Hence in real world it won't give meaningful results.

What's needed is some kind of regularization.

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