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.

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