Suppose the following classical supervised regression setting, $$y_{i} = f(x_{i}) + \epsilon_{i}, \quad i=1,\cdots,n,$$ where $\epsilon_{i}$ are i.i.d. zero mean Gaussian noise.
The above regression problem can be solved by classical non-parametric approaches, e.g. kernel method.
However, if I suppose that I have a pre-determined (or pre-trained) model $h(x)$, such that $$f(x) = g(x,h(x)).$$ Here, the $g$ is a bivariate function but with the second coordinate restricted in the form of $h(x)$.
When none of the variables is restricted, one can apply the kernel method (like kernel ridge regression) to obtain the estimate $\hat{g}$ in the tensor space of the RKHS. However, I'm wondering if there is any existing literature/research for estimating $g$ given $g$ in such a restricted form.