用于单阶段哈达玛量化的两两独立抖动
Pairwise-Independent Dithering for Single-Stage Hadamard Quantization
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中文总结 AI 辅助
针对高维向量量化的两阶段哈达玛量化方法存在通信开销大、主导常数高的问题，提出了基于两两独立抖动的单阶段哈达玛量化方法，消除了残差阶段，降低了约5.93倍的主导常数，相关证明由谷歌Gemini智能体系统完成并验证。
中文摘要 AI 辅助
对高维向量进行量化是相似性搜索、分布式学习和模型压缩的基础。Feng、Indyk、Kapralov、Krachun和Prokhorov针对基于随机哈达玛变换的无偏抖动量化器建立了严格的保证[FIK+26]。然而，他们的1/d尺度内积估计器使用了第二个随机变换和残差量化，这既增加了通信开销，也提高了已证明界限中的主导常数。我们证明了这个额外阶段是不必要的：跨哈达玛坐标的两两独立抖动就足够了。得到的无偏单阶段估计器每个坐标使用b比特，当b→∞时，满足E[|⟨y,x̂−x⟩|²] ≤ (3π√3/2 + o(1)) ||y||₂²/(d·4ᵇ)，其中o(1)项与维度无关，在单位输入和固定查询上均匀。与Feng等人的两阶段构造相比，它消除了残差阶段O(d)比特的有效载荷，并将主导上界常数降低了约5.93倍。该证明最初是使用谷歌内部开发的基于Gemini的全自动智能体系统获得的，作者已验证该证明并对其进行了编辑以提高表述清晰度。
英文摘要
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.
