Why does data get so tangled up in high dimension? When I look at textbooks on classification and machine learning, many of the examples focus on data that is often twisted up such as to avoid linear separation. I have an example picture below. The common description of the classification problem is that data can be twisted in this manner, and hence kernel methods or random forests are better at dealing with nonlinear decision boundaries. Of course, there is no a priori reason that datapoints should be nicely separated and linearly separable.

But I was trying to understand how the data usually gets so twisted up? First, the usual claim is that in high dimension, meaning perhaps dimensions greater than 10 or so, the distance between points becomes so big, that all points are essentially the same distance from each other. If that is the case, then I would imagine that data should not be twisted, because all the points are far from both similar points and dissimilar points.
At the same time, there is this common manifold assumption, where data generally lie on a lower dimensional manifold. But again, how would this manifold assumption reconcile with the distance issues in high dimension? As we add dimensions, points seem to become more equally distant from each other.
So I am just trying to understand how the geometry of high dimensional data evolves, given the contrast between the high dimensional distance issues versus the manifold assumption?

Textbook examples are not meant to represent the likelihood of encountering particular situations. They are meant to cover a wide range, and give you the ability to recognize and solve certain special situations even if you rarely encounter it in practice.

When I look at textbooks on classification and machine learning, many of the examples focus on data that is often twisted up such as to avoid linear separation. I have an example picture below. The common description of the classification problem is that data can be twisted in this manner, and hence kernel methods or random forests are better at dealing with nonlinear decision boundaries. Of course, there is no a priori reason that datapoints should be nicely separated and linearly separable.

IMHO the keyword here is can.
I.e., "if your data looks like this, [linear won't work but] you can do as follows..."
In general, data can be whatever from easily linearly separable over complicated as in the illustration to truly inseparable.
However, we can make statements how typical data of certain types/domains/applications/tasks looks.
"My" data are mostly vibrational spectra.

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*For them, the physics and chemistry behind the data-generation processes mean for particular questions like medical differential diagnosis that chances are quite good that a linear separation (or at least not too complex non-linear) can be found.
In practical terms it behaves like you reason: since we typically cannot get exponentially more cases (sample size) with increasing dimensionality, the sampling density will be lower and chances are that data are less tangled. Of course, estimating class boundaries is also more uncertain because of the lower sampling density.
So it may be linearly separable in the original (≈ 1000d) space, but compressed into low dimensionality (say, 10d), class boundaries may become nonlinear.
BTW, it is typically not clear beforehand whether modeling in original or low-d projection will work better.


*For other tasks such as quality control where "good" means being not too far away from the target, a (one) linear separation cannot tackle too low and too high at the same time. The data would be expected to behave more like the donut, but without a low-density region between good and bad and maybe with additional red clusters far away.
And this is expected to be true both in high dimensional original space as well as in low-d projections of it.
