# Solving a system of nonlinear equations involving only products of unknowns

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?

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.