Fractal is not necessarily self-similar; it just has to show enough detailed structures when zoomed in ad infinitum. While no single defiition for fractals exists, at least that's one of the common denominators (because otherwise many "fractals" in the nature can't be called so).

I'm only a fractal enthusiast, but my impression is that the key distinction that makes these fractals, or at least fractal-like, is not their detail per se, but that there is complexity at every scale. From the article:

> we find intricate structure at every scale

> At every length scale, small changes in the hyperparameters can lead to large changes in training dynamics

"small changes in the hyperparameters can lead to large changes in training dynamics"

This is the definition of Chaos though no? Butterfly flaps its wings, hurricane on other side of the planet...

As far as I am aware, these kinds of nonlinear relationships are a feature of fractal dynamics, but I'm not a mathematician.

It would be interesting to compute the fractal dimension[1] to see if it really is fractal[2] or just looks like it.

I recall similar tests being done on paintings by Pollock and of other artists trying to copy his style to determine authenticity[3].

[1]: https://en.wikipedia.org/wiki/Fractal_dimension

[2]: https://mathworld.wolfram.com/Fractal.html

[3]: https://cpb-us-e1.wpmucdn.com/blogs.uoregon.edu/dist/e/12535...

The sister paper[1] contains the fractal dimensions of some of the images using a boxcount paper.

[1]: https://arxiv.org/pdf/2402.06184.pdf

Ah I had a quick read before I went to bed last night, missed that bit, cheers.

Thanks!! I was also under the impression Fractals required self similarity and not just infinite detail but glad to have that misconception corrected!

I think it probably stems from the fact that all the fractals, I have seen, for which the dimension can be analytically calculated do show obvious patterns of similarity at different scales.

Searching I stumbled over this paper[1], which I found interesting, on characterizing the fractal-ness of geological structures.

They point out that the usual measure of fractal dimension, or capacity dimension[2], doesn't consider the physical size of the features and can thus be inaccurate. Instead they suggest using the information dimension[3], which is bounded by the capacity dimension.

[1]: https://doi.org/10.1016/j.chaos.2018.05.008 (full text available on that hub of science)

[2]: https://mathworld.wolfram.com/CapacityDimension.html

[3]: https://mathworld.wolfram.com/InformationDimension.html

Seems there are a few more, Higuchi Dimension, Box Counting Dimension (D0), Generalized Rényi Dimension and others are discussed mid-way through this;

https://medium.com/@h.a.papageorgiou/the-reality-of-the-ruli...