This subject has probably been studied before, but my search skills are failing me right now.
Suppose I have a nonlinear function $f$ of many variables $x_1, ..., x_N$, forming a vector $\mathbf{x}$ which lies in a hyperrectangular set $X = [0,1]^N \in \mathbb{R}^N$.
I wish to find another function $g$ of variables $y_1, ..., y_M$ with $M<N$ such that for any $\mathbf{x}$ there is a $\mathbf{y}$ that makes $f(\mathbf{x})$ and $g(\mathbf{y})$ "as close as possible".
For example, a hypothetical method could find, at the same time as it finds $g$, a mapping $\mathbf{y} = \alpha(\mathbf{x})$ such that the following L2(?) cost is minimised
$$ \min_{g,\alpha} \int_X \lVert f(\mathbf{x}) - g(\alpha(\mathbf{x})) \rVert^2 \mathrm{d} \mathbf{x} $$
Note: Ideally the set containing $\alpha(\mathbf{x})$ is as simple as possible.
Note2: I realise as I write this that the mapping $\alpha$ might be all I need.
Is there any book or paper or known methodology handling this problem?
P.S.: In my research, I found the following paper with a promising title: Golomb M, Approximation by functions of fewer variables, On numerical approximation, Proceedings of a Symposium, Madison 1959, edited by R E Langer (The University of Wisconsin Press) pp. 275–327
But I can't access it, and other works that cite it use advanced mathematical language that takes me too much time to understand and seems too abstract to lead to an actual numeric implementation of a method.
Also, searches for "dimensionality reduction" lead to methods that find a latent space of some dataset, but that doesn't apply here. The variables $\mathbf{x}$ can be anywhere within the N dimensional set $X$.