I wanted to create a function that its shape was a function of the depth of the compositionality (on a fixed interval). For example consider some compositional function
$$f(x_1, x_2) = g( g( g( h_1(x_1,x_2) ) ) )$$
ideally, when $g$ is applied more times, I'd like the visual shape of the function to change in a way that is noticeable and distinguishable compared to applying it less times (i.e. the compositionality is the important factor).
I tried the function:
$$ f([x_1,x_2]) = g_L\circ...\circ g_1( cos( cx_1 x_2 ) ) = \bigcirc^L _l g_l( cos( cx_1 x_2 ) ) $$
where $g_l(x) = g(x) = k(ax^2+bx_1+c)^p$ (with k=0.5, a=1, b=1.01, c=1.1, p=0.6). When I tried three layers of compositional depth (i.e. L=3,2,1) I got the following three graphs:
Unfortunately, it seems that its essentially the same graph just sort of re-scaled but it doesn't seem any different in any interesting way. I tried other more complicated function but got similar behaviour in that it looks more complicated but adding more depth to the compositionally didn't seem to change the function in any noticeable way (at least not when inspected visually). Other issues that I've found was that adding more compositionally seemed to explode or vanish some regions of the function. I guess that is changing the function a bit but it seemed like boring properties that in the end (form my observations) essentially just smoothed out the curve.
Of course, from my description I have not thought of a metric to measure this type of difference (or do I require it to solve my problem), but it would be nice to have a function that is sort of recursive/compositional (in 2D) that doesn't just rescale the same picture when added with more compositional depth.