Binary quantization (BQ) compresses high-dimensional embeddings into one or two bits per coordinate, enabling nearest neighbor search at extreme speed. Yet a striking puzzle persists: BQ achieves competitive recall on contrastive embeddings but fails on others -- and two leading systems adopt diametrically opposite strategies (random rotation vs. preserving coordinate axes) without a common theory explaining when each is appropriate. We address this puzzle by connecting the Gaussian structure recently established for InfoNCE-trained representations to a statistical framework for BQ quality. Our analysis reveals two distinct roles of the covariance matrix. First, the full covariance structure -- not merely its diagonal -- determines the absolute level of ranking fidelity, with off-diagonal correlations contributing 30--50% of the signal. Second, coordinate heterogeneity (the non-uniformity of per-coordinate variances) governs key design choices: how much each additional bit contributes, and whether random rotation helps or hurts. We derive approximate expressions for ranking fidelity under a Gaussian model, show that the magnitude bit carries information proportional to heterogeneity, and show that random rotation destroys precisely the signal that one paradigm exploits while creating the isotropy that the other requires. A phenomenological scaling law predicts fidelity across models and dimensions. Experiments on 18 datasets spanning 9 embedding families support the main predictions and provide, to our knowledge, the first principled design guide for binary quantization systems.

翻译：摘要：二值量化（Binary Quantization, BQ）将高维嵌入压缩至每坐标一或两比特，从而以极高速度实现最近邻搜索。然而，一个显著谜题始终存在：BQ在对比嵌入上取得具有竞争力的召回率，却在其他嵌入上表现不佳——且两种主流系统采取截然相反的策略（随机旋转vs.保留坐标轴），却缺乏统一理论解释各自适用场景。我们通过将近期建立的InfoNCE训练表示的 Gaussian 结构与BQ质量的统计框架相联系，破解了这一谜题。分析揭示了协方差矩阵的双重作用：首先，完整协方差结构（而非仅其对角元）决定了排序保真度的绝对水平，其中非对角相关性贡献了30-50%的信号；其次，坐标异质性（各坐标方差的非均匀性）主导关键设计选择：每个额外比特的贡献度以及随机旋转的利弊效果。我们推导出Gaussian模型下排序保真度的近似表达式，证明幅度比特携带与异质性成正比的信息，并表明随机旋转恰好破坏了一个范式所依赖的信号，同时创造了另一范式所需的各向同性。所提出的现象学标度律可跨模型与维度预测保真度。在覆盖9类嵌入家族的18个数据集上的实验结果支持了主要预测，并据我们所知，首次为二值量化系统提供了基于原理的设计指南。