Concentration of measure, near-orthogonality of random vectors, the curse — and how rotation fights it.
In high dimensions, the corners of a hypercube swallow all the volume and pairwise distances stop discriminating between near and far. Both phenomena make exact nearest-neighbour hopeless above d ≈ 30 — and are the reason approximate NN is the real game.
In high dimensions, smooth functions of many independent variables stop varying. Standard Gaussian samples concentrate on a thin shell of radius √d. The fact that's so weird — and that *rescues* high-D ML from the curse.
Two random unit vectors in ℝᵈ have dot product ⟨u, v⟩ ≈ N(0, 1/d) — concentrated near 0. You can pack exp(c·d) vectors that are all pairwise "almost orthogonal" — exponential capacity. This is why high-D embeddings work, why JL works, and why TurboQuant's rotation step is structurally a JL transform.
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