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Comment by david-gpu | Hacker News Reader

> For example high dimensional spheres are "pointy" and most of their volume are near their surface

I had a visceral reaction to this. In what sense can a sphere be considered pointy? Almost by definition, it is the volume that minimizes surface area, in any number of dimensions.

I can see how in higher dimensions e.g. a hypersphere has much lower volume than a hypercube. But that's not because the hypersphere became pointy, it's because the corners of the hypercube are increasingly more voluminous relative to the volume of the hypersphere, right?

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There is a standard thought experiment where you start with a hypercube of side-length 2, centered at the origin. You then place a radius 1 sphere on each vertex of this hypercube. The question then becomes: what is the largest sphere you can place at the origin so that it is "contained" by the other spheres. As it turns out in like dimension 6 or so the radius of the center sphere exceeds 1. It will actually poke out arbitrarily far (while still being restricted by the corner spheres).

Yes, but that can be better understood as the hypercube becoming more pointy, not the sphere. And it's true; the cube's vertices get arbitrarily far from the origin, while the centers of its faces stay at ±1.

There are other ways in which a hypersphere can be considered "pointy", though; for example, consider a point lying on the surface being moved some epsilon distance to a random direction. As the dimension increases, the probability that the point ends up inside the sphere approaches zero – the sphere spans a smaller and smaller fraction of the "sky".

Specifically, of course, d = sqrt(N), where N is dimension and d is distance of a vertex of the unit hypercube from the origin.

I hear this point parroted all of the time, but I think it is a misunderstanding and a poor visualization. Consider the same situation, but instead of focusing on the radius of the center sphere, focus on the distance between the spheres on the corners to the origin. For 1-dimension, these 'spheres' are unit intervals and so the distance is 1 (Central radius is 0). For 2-dimensions, these are circles at a distance of root(3) (Central radius is root(2)-1). 3-D: root(3) (Central radius is root(3)-1). Etc. So, it isn't the central circle getting more 'pointy' allowing the central radius to increase, but rather that the corner circles are getting further from the origin, allowing larger N-spheres (increasing proportional to the root of N). Thus, pointy is not the right way to conceptualize these spheres. For the more visual folk, I would recommend drawing this out and you can see this in action. More clearly, if a sphere became 'spikey' then the distance on the surface of the spike should be further than a neighboring point, which is NOT the case. Not trying to attack you, I just see this same point over and over and think that this warrants more thought

Yes that is true, but there are other ways to see spiky-like behavior.

First, the volume of spheres (or balls rather) in higher dimensions goes to zero as the dimension grows. Said another way, to keep unit volume on a ball you need to grow the radius more and more (which I interpret as spiky).

Second, the volume of spherical caps grows like ~exp(- d h^2 /2), in particular the caps lose volume fast in higher dimensions. To interpret this as "spikyness" I like to visualize it as two balls intersecting (which is just 2x the cap volume). If they are of the same radius, but their centers are just slightly off their intersection volume goes to zero quickly!

https://news.ycombinator.com/item?id=3995615 (both article and comments) describe various ways of looking at this - and there are many implications for machine learning e.g. https://news.ycombinator.com/item?id=3995964 !

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