Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Counterintuitive Properties of High Dimensional Space

marckhoury.github.io

Post reply on HN
Print view View original

41–50 of 68 posts

Re: Counterintuitive Properties of High Dimensional Space

#41

Post by yequalsx »

ttoinou wrote: post #40
DavidSJ wrote: post #37

Earlier quoted context omitted.

I think we’re not really disagreeing about anything. In a pure math context, x would be, as you say, a quantity in which dimensions play no role.

As he said, x is part of a ring. In physics the multiplication of two elements from the same ring end up on another ring (meters^2 for example) in which, for example, you cannot do addition with elements from the original ring (meters)

The mathematical definition of a ring precludes this from happening. By definition, in a ring when you multiply 2 elements from the ring you get another element of the ring. Perhaps you are referring tensor products (geometric algebra for physicists).

Re: Counterintuitive Properties of High Dimensional Space

#42

Post by signa11 »

btilly wrote: post #13

Anyone interested in practical consequences of these counterintuitive properties should look up https://en.wikipedia.org/wiki/Curse_of_dimensionality for how it impacts things like machine learning where we naturally are working with many dimensions.

indeed ! the 'neural networks for pattern recognition' book (bishop) has exercises in chapter-01 on the same subject.

Re: Counterintuitive Properties of High Dimensional Space

#44

Post by John_KZ »

resource0x wrote: post #10

from the article: The volume of the unit d-sphere goes to 0 as d grows! A high dimensional unit sphere encloses almost no volume! Maybe I'm nitpicking, but this interpretation is not accurate IMO. Volume of N-dimensional sphere is measured is different units than that of (N-1)-dimensional sphere. E.g. one is m^5, and another is m^4. Comparing values measured in different units is not the best idea. It would be better…

Exactly. The article is really bad and serves only to confuse those with no experience in high-dimensional mathematics.

Re: Counterintuitive Properties of High Dimensional Space

#45

Post by [deleted] »

resource0x wrote: post #10

from the article: The volume of the unit d-sphere goes to 0 as d grows! A high dimensional unit sphere encloses almost no volume! Maybe I'm nitpicking, but this interpretation is not accurate IMO. Volume of N-dimensional sphere is measured is different units than that of (N-1)-dimensional sphere. E.g. one is m^5, and another is m^4. Comparing values measured in different units is not the best idea. It would be better…

[deleted]

Re: Counterintuitive Properties of High Dimensional Space

#47

Post by adrianratnapala »

resource0x wrote: post #10

from the article: The volume of the unit d-sphere goes to 0 as d grows! A high dimensional unit sphere encloses almost no volume! Maybe I'm nitpicking, but this interpretation is not accurate IMO. Volume of N-dimensional sphere is measured is different units than that of (N-1)-dimensional sphere. E.g. one is m^5, and another is m^4. Comparing values measured in different units is not the best idea. It would be better…

Yes. But I've learned from bitter experience that mathematicians don't believe in units. At least not in their gut. To them, it's all just numbers.

(Even more infuriating is that physicists do believe in units, except when writing code).

Re: Counterintuitive Properties of High Dimensional Space

#48

Post by qubex »

antognini wrote: post #18
chatmasta wrote: post #15

Why do we extrapolate 4d based on 3D volume, and not surface area? For 2D->3D we use area. Why would we then use volume in 3D->4D when surface area is an option?

We're using volume the whole time. Area is just the two-dimensional equivalent of volume. The two-dimensional equivalent of surface area would be perimeter, which is relatively rarely used.

The more general term for ’area’ and ’volume’ in higher dimensions is ’content’: area is the content of a 2D item, volume is the content of a 3D item, hypervolume is the content of a 4D item, et cetera.

Re: Counterintuitive Properties of High Dimensional Space

#49

Post by qubex »

yequalsx wrote: post #33
chatmasta wrote: post #15

Why do we extrapolate 4d based on 3D volume, and not surface area? For 2D->3D we use area. Why would we then use volume in 3D->4D when surface area is an option?

The space that is enclosed by a unit sphere in dimension 1 is the interval (-1, 1). In physics this is measured in meters. This is referred to as length. The space enclosed by the unit sphere in dimension 2 is measured in meters squared. This is referred to as area. In dimension 3 we call it volume and measure it in meters cubed. In dimensions 4 through infinity we quickly come to a problem. Do we come up with unique…

In my applied mathematics education the highest-dimensional unit of measurement of a given dimensionality was referred to generally as ‘content’: length is the content of 1D objects, area is the content of 2D objects, volume is the content of 3D objects, hypervolume is the content of 4D objects, and henceforth you just start numbering them...

Re: Counterintuitive Properties of High Dimensional Space

#50

Post by visarga »

An interesting property of word vectors (which are usually 300-600 dimensional vectors) is that most are quasi-orthogonal. That means when you sum them up, they compose well representing all the meanings of the component parts, and strangely, multi-sense words such as "bank" contain all the senses overlapped, yet distinct.

Another interesting property is that high dimensional space has many shortcuts, or that at any point there are many paths. It's like a kaleidoscope with infinite reflections or like a mirror house. Or it's like any point has many close neighbours which can, paradoxically, be far apart between them.

Post reply on HN