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.

table::

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

code: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 x0 > 0.2:

over020 += 1

if x0 > 0.22:

over022 += 1

if x0 > 0.24:

over024 += 1

print(dim, over020, over022, over024)

higher dimensional space.

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