Paper Detail
Honghao Lin, Vahab Mirrokni, David P. Woodruff
Quantizing high-dimensional vectors is fundamental to similarity search, distributed learning, and model compression. Feng, Indyk, Kapralov, Krachun, and Prokhorov established sharp guarantees for an unbiased dithered quantizer based on a randomized Hadamard transform [FIK+26]. Their $1/d$-scale inner-product estimator, however, uses a second randomized transform and residual quantization, increasing both communication and the leading constant in the proved bound. We show that this extra stage is unnecessary: pairwise-independent dithers across Hadamard coordinates suffice. The resulting unbiased single-stage estimator uses $b$ bits per coordinate and achieves \[ \mathbb{E}\!\left[ \left|\left\langle y,\widehat{x}-x\right\rangle\right|^2 \right] \leq \left(\frac{3π\sqrt{3}}{2}+o(1)\right) \frac{\lVert y\rVert_2^2}{d\,4^b}, \] as $b\to\infty$, with a dimension-free $o(1)$ term uniform over unit inputs and fixed queries. Compared with the two-stage construction of Feng et al., it eliminates the residual-stage $O(d)$-bit payload and reduces the leading upper-bound constant by a factor of approximately $5.93$. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.
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@article{lin2026pairwise,
title = {Pairwise-Independent Dithering for Single-Stage Hadamard Quantization},
author = {Honghao Lin and Vahab Mirrokni and David P. Woodruff},
year = {2026},
abstract = {Quantizing high-dimensional vectors is fundamental to similarity search, distributed learning, and model compression. Feng, Indyk, Kapralov, Krachun, and Prokhorov established sharp guarantees for an unbiased dithered quantizer based on a randomized Hadamard transform [FIK+26]. Their \$1/d\$-scale inner-product estimator, however, uses a second randomized transform and residual quantization, increasing both communication and the leading constant in the proved bound. We show that this extra stage i},
url = {https://arxiv.org/abs/2608.02564},
keywords = {cs.DS},
eprint = {2608.02564},
archiveprefix = {arXiv},
}
{}