On: Counterintuitive Properties of High Dimensional Sp...

[kgwgk]

> In dimensions 1, 2 and 3 these look like a fuzzy ball around the origin. But in high dimensions, it is more like a thin shell of radius sqrt(n).

But it’s also still like a ball. The “shell” thing is not something particular the Gaussian: the same happens for a hard ball. As the dimension increases the share of the mass of the ball close to its surface goes up.

[rich_sasha → kgwgk]

That's true for any ball, even in 2D - most volume is by the edge. But a Gaussian in low dimensions looks "closer" to 0.

[kgwgk → rich_sasha]

Sure, in a 2D ball most volume is by the edge and most of a 2D Gaussian is around radious sqrt(2).

In either case the “concentration” gets more and more important as the number of dimensions goes up.

(Concentration in quotes because for the Gaussian the typical density goes down.)