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?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
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.
