Simple approximation for spherical cap area

The previous post looked at how to interpret cosine similarity, or equivalently angles between word vectors. In a high-dimensional space, randomly chosen vectors are likely nearly perpendicular, and so relatively large angles, such as 50, indicate very closely related words.

Another way to look at this, as explained in the previous post, is that in high dimensions, a spherical cap of angular radius represents a small portion of a sphere, even for moderately large .

The proportion of the area inside the spherical cap, given here, involves the regularized incomplete beta function" and so it's hard to have an intuition for the value.

For large dimension n, the approximation

n^-1/2 sin^{n - 1}()

gives the proportion of the area inside the cap to within an order of magnitude. It's easy to see that this function goes to zero quickly asn increases, provided || < /2.

If you have the cosine similarityc = cos rather than itself, the approximation becomes

n^-1/2 (1 - c^2)^{(n - 1)/2}.

Python script

Let's try it on the example from the previous post, in whichn = 200 and = 49.

import numpy as npfrom scipy.special import betainc# Fraction of S^{n-1} inside a spherical cap of angular radius theta# theta is measured from the pole# Assume 0 < theta < pi/2def cap_fraction(theta, n): x = np.sin(theta) ** 2 return 0.5 * betainc(0.5 * (n - 1), 0.5, x)def cap_fraction_approx(theta, n): return n**(-0.5) * np.sin(theta)**(n-1)theta = np.deg2rad(49)print(cap_fraction(theta, 200)) print(cap_fraction_approx(theta, 200)) 

This prints 2.03e-26 and 3.37e-26. The order of magnitude is correct as advertised.

The post Simple approximation for spherical cap area first appeared on John D. Cook.