Dithered quantization after a single randomized Hadamard transform yields unbiased estimates whose MSE asymptotically equals that of dense random rotations, specifically (π√3/2 + o(1))·4^{-b} for b-bit TurboQuant.
Fast Cross-Polytope Locality-Sensitive Hashing
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abstract
We provide a variant of cross-polytope locality sensitive hashing with respect to angular distance which is provably optimal in asymptotic sensitivity and enjoys $\mathcal{O}(d \ln d )$ hash computation time. Building on a recent result (by Andoni, Indyk, Laarhoven, Razenshteyn, Schmidt, 2015), we show that optimal asymptotic sensitivity for cross-polytope LSH is retained even when the dense Gaussian matrix is replaced by a fast Johnson-Lindenstrauss transform followed by discrete pseudo-rotation, reducing the hash computation time from $\mathcal{O}(d^2)$ to $\mathcal{O}(d \ln d )$. Moreover, our scheme achieves the optimal rate of convergence for sensitivity. By incorporating a low-randomness Johnson-Lindenstrauss transform, our scheme can be modified to require only $\mathcal{O}(\ln^9(d))$ random bits
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Provable Quantization with Randomized Hadamard Transform
Dithered quantization after a single randomized Hadamard transform yields unbiased estimates whose MSE asymptotically equals that of dense random rotations, specifically (π√3/2 + o(1))·4^{-b} for b-bit TurboQuant.