Cosine similarity of 0.2 is extremely rare in higher dimensions

In high dimensions of about 400 dimensions, Cosine similarity 0.2 is extremely rare, 1 in 10,000 times.

• I really wanted to try it in 768 dimensions, but after a million tries I got zero. :

  dim	over 0.20	over 0.22	over 0.24
  2	4297	4245	4177
  10	2760	2554	2379
  100	228	140	76
  200	25	9	1
  300	5	4	1
  400	1	0	0

py

import numpy as np
for dim in [2, 10, 100, 200, 300, 400]:
    over020 = 0
    over022 = 0
    over024 = 0
    for i in range(10000):
        x = np.random.randn(dim)
        x = x / np.linalg.norm(x)
        if x[0] > 0.2:
            over020 += 1
        if x[0] > 0.22:
            over022 += 1
        if x[0] > 0.24:
            over024 += 1

    print(dim, over020, over022, over024)

higher dimensional space.

This page is auto-translated from /nishio/高次元においてコサイン類似度0.2は激レア using DeepL. If you looks something interesting but the auto-translated English is not good enough to understand it, feel free to let me know at @nishio_en. I'm very happy to spread my thought to non-Japanese readers.