Two-block Hadamard rotations match uniform rotations in each coordinate with improving accuracy as dimension grows but differ substantially in their overall distribution, with explicit bounds.
Yu, Ananda Theertha Suresh, Krzysztof Choromanski, Daniel Holtmann-Rice, and Sanjiv Kumar
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Approximating Uniform Random Rotations by Two-Block Structured Hadamard Rotations in High Dimensions
Two-block Hadamard rotations match uniform rotations in each coordinate with improving accuracy as dimension grows but differ substantially in their overall distribution, with explicit bounds.