Dimension d = 3 Input vector

mean angle to random rows

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std dev of angles

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max |new coord|

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std dev of new coords

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angle between x and each row of Π

In high-d, all angles cluster near 90°

new coordinate values (= dot products with rows)

All similar magnitude because all angles ≈ 90°

original coordinates after rotation (= dot products with random dirs)

Before vs after: spiky becomes uniform because each coord = ⟨row_i, x⟩ ≈ cos(~90°)

angle std dev (decreasing = more concentrated at 90°)

Angle concentration vs dimension: higher d → angles pack tighter around 90°