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I would like to find the solution of a system of equations of the form:

$A = w_1 F(x_k) + w_2 F(x_l) + w_3 F(x_m) +...$,

where the unknowns are the $w_i$ and the function $F$, while my data are the $A$ and $x_i$s. I have a system of those equations for different $A$s and different points $x_i$ where the function $F$ is evaluated.

The plan is to expand the function $F$ on some basis.

I am wondering whether there is some numerical method specifically designed for problems with this kind of structure.

One possible solution would involve iterations with an alternating least-squares between optimizing the $w_i$s while holding the $F$ fixed and then optimising the function approximation for fixed $w_i$s. Yet I am not sure whether this approach would converge.

Is anyone of you aware of something along these lines?

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The equation described here looks familiar to many machine learning approaches.

It appears to fit the paradigm of a regression equation where you have designed/engineered "features" by applying a function, $F$, to your raw data, $x_i$.

The $w_i$'s would be the weights that can be learned by least squares or other methods.

If you re-use the function $F$, this problem could be structured as a single hidden-layer neural network. In this case $F$ would typically be a function of the group called activation functions.

It would be ideal to explore the addition of an intercept/bias term to the equation for A.

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