Earlier quoted context omitted.
I took the author's use of O(n) vs O(n^2) as a framing point rather than a literal model. It just seems to be missing the forest for the trees. Besides, we can approximate sigmoids with linear or quadratic functions when windowing them. Considering startup as context I think we know what part of the graph we're talking about... Do we see that exponential explosion or is the sigmoid much more flat. Replace the x in yo…
That’s an interesting thought, about windowing. I think it is clear that O(n) vs O(n^2) is really just an analogy so we shouldn’t over-formalize it. But it is interesting to note that a sigmoid could be thought of as looking like either one, depending on when you look at it. It makes me wonder if there’s some sampling bias that is accidentally being applied. Because, another way of looking at it could be: assume I ob…
For wondering, I believe this might help. If we parametrize sigmoid to be 1/(1+e^{-ax}) then our a factor is at play. We're interested in xI still think the model is naive. Even just a sigmoid is. You need the composition of them and further parameterization. They are good for approximating as you originally suggested but this is similar to using fourier series. It's the fact that you have the three classes of second derivatives allowing you to strongly control the composite function through minimal parameterization and complexity. But that's also why I immediately understood this to be being using in an illustrative way rather than a serious one. I mean realistically so are you.