Hacker News Reader

Comment by shenberg | Hacker News Reader

The spiky view of n-spheres is due to all of their volume getting concentrate next to the axes - the proportion of volume which is less than ε away from an axis to the entire volume of the sphere tends to 1 exponentially with n, which leads to the spiky visualization as we imagine 3d but with more axes.

View on HN

I don't think this is the right way to process this intuitively. When you use words like "volume getting concentrated" it sounds like there is some non-uniformity in the sphere, but the non-uniformity is really in our intuition about space.

What's weird isn't the sphere, it's distance, and I think that's easier to process. Going from a (1d) sidewalk to a (2d) football field to a (3d) ocean, it's easier to see our intuitions about distances slowly breaking down.

>it sounds like there is some non-uniformity in the sphere

How I am understanding this is that the non-uniformity is in the cube, and I think it's very helpful in visualizing it.

I can imagine in 3d, the centers of the sides of the cube being pulled in, so that it's kind of hyperbolic looking.

That's not accurate either. Neither the cube nor the sphere "bend" in any way. Irrespective of the number of dimensions, the cube's surfaces remain (hyper)planar, and the sphere retains its convex shape and constant curvature.

The diagram on the web page is just wrong. The enclosed sphere does not poke out with "spikes".

The error in the diagram is that it shows the corner cubes "filled" with spheres, but this is not what happens in the higher-dimensional analogues. If you take a cube-shaped cross section of a 4D example (through the center), you wouldn't see the corner spheres at all.

Think of a 2D slice through the middle of the 3D example: You'd just see a square, no circles!

>If you take a cube-shaped cross section of a 4D example (through the center), you wouldn't see the corner spheres at all.

Wait wait wait, I thought the "cube" had the same number of dimensions as the "spheres".

They're the same. So square + circle, cube + sphere, 4D hypercube + 4D hypersphere, etc...

But cross-sections don't look the way you think they do. So if you draw a 3D intersection through the 4D case, it looks much like the 2D intersection of the 3D case, etc...

In all such "intersection" diagrams you don't see the spheres in the corners of the cubes. The conceptual diagram on that page showing the high-dimensional case shows the corner spheres and the middle sphere somehow "squished out" in a spiky way.

This is not at all what happens.

I'm not certain I know what you mean, but it doesn't seem to work. If you mean to take epsilon neighborhoods in Euclidean norm of the axes, then I'm skeptical that that contains most of the mass of the sphere, since simulating random points on a thousand dimensional sphere doesn't seem to give points near these axes.

If instead you mean most of the mass of the ball is in the points which have all but one coordinate within epsilon of zero, then the intuition doesn't follow. It's equally true that you get most of the mass considering just points with all coordinates within epsilon. And for that you get that the mass of the ball is concentrated in an epsilon cube at the origin, which excludes exactly the spikes that you were basing the intuition on. The weird thing in this thought experiment is the epsilon cube not the sphere. For example the epsilon cube contains points much further than epsilon from the origin and so it's maybe not surprising that it contains most of the sphere.

https://news.ycombinator.com/item?id=3995615

Perhaps you can be more specific since I don't see the claim here being discussed there.

>since simulating random points on a thousand dimensional sphere doesn't seem to give points near these axes.

that is kind of circular argument as "random" really depends on the density measure underlying the chosen sampling distribution.

It's no more ill defined than volume itself (and less so since it doesn't need an arbitrary scale).

I don't understand your 'less so' comment. Could you please elaborate?

If you want to define volume in D-dimensional space then you need to choose what the unit volume is, for example that a unit hyper cube has unit volume one. (That choice is relevant for these discussions and you could equally have chosen a unit ball to have volume one since you're defining D-dimensional volume you get to choose.) But a probability distribution is always normalized to integrate to one, which means either choice of volume definition will give you the same uniform probability distribution on the ball. So there's an extra arbitrary constant to choose for volume that isn't important in the probability distribution even though you define the probability density in terms of the volume.

OK. Though you could leave your volume definition parameterized, instead of picking a specific scale, couldn't you?

I do agree that volume and probability distribution are essentially the same.

(That's also why at my university measure theory and probability theory were handled in the same course.)

If only there was some kind of natural density measure to use in an Euclidean space R^D…

The n-cube is spiky. The n-sphere is kinda defined by its non-spikiness, because it is rotationally invariant.

But there is an alternative visualization where the sides of the unit (hyper)cube are deformed because the location of the vertices gets further and further away from the origin. Of course neither is a true representation of the hyperdimensional case.

Love this explanation, visualizing corner vs centre is intuitive in a way higher dimensions usually aren't.

Reply on news.ycombinator.com