
Intuitive crutches for higher dimensional thinking - saeranv
https://mathoverflow.net/questions/25983/intuitive-crutches-for-higher-dimensional-thinking/26095#26095
======
saeranv
Terence Tao's answer here fascinates me, even though I only partially
understand his response.

> For instance, one can view a high-dimensional vector space as a state space
> for a system with many degrees of freedom. A megapixel image, for instance,
> is a point in a million-dimensional vector space; by varying the image, one
> can explore the space, and various subsets of this space correspond to
> various classes of images.

I don't understand this. How is he thinking of a 2D data structure as a n-dim
vector space? The only way I can think of mapping n-dims to 2d is maybe taking
two n-dimensional vectors and visualizing the 2D plane between them, or
projecting n-dimensions onto 2D space...

> For instance, the fact that most of the mass of a unit ball in high
> dimensions lurks near the boundary of the ball can be interpreted as a
> manifestation of the law of large numbers, using the interpretation of a
> high-dimensional vector space as the state space for a large number of
> trials of a random variable.

I think this came up recently in another hnews thread. The way I interpret
this statement is that you can interpret the probability of events occurring
in a portion/segment of a n-dimensional space as permutations of those
segmented events in each dimension:

1/k^n where: k = number of segments n = number of dimensions

So for example, if you have a 1D line, and you want to find the probability of
ending up in a third of the line, the formula is: 1/3^1 = 0.333

If we think extend this to a plane, and look at the probability of ending up
in the third of 2 dimensions:

1/3^2 = 1/9

Would be curious if someone has another interpretation.

