极值顺序统计量伴随的次线性最大内积搜索
Sublinear Maximum Inner Product Search using Concomitants of Extreme Order Statistics
摘要：
基于极值顺序统计量伴随理论，提出了一种新的最大内积搜索（MIPS）的降维方法，称为CEOs。利用这些伴随的渐近行为，我们表明，与查询签名的极值相关联的几个维度足以估计内积。由于CEO只使用查询签名的一个小子集的符号进行估计，我们可以在查询之前准确地预先计算所有内积估计量。这些性质产生具有指数索引空间复杂度的次线性MIPS算法。我们表明，我们的指数空间是最佳的$(1 + ε)$ -近似MIPS上的单位球。在一定的条件下，CEO的搜索召回率在理论上是可以保证的。为了处理指数空间复杂度，我们提出了两个实用的变体，包括sCEOs-TA和coCEOs，使用线性空间求解MIPS。sCEOs-TA利用阈值算法（TA），并提供卓越的搜索召回竞争MIPS求解器。coCEOs是一种数据和维度共约简技术，在高召回要求上优于sCEOs-TA。从经验上讲，它们非常容易实现，与蛮力搜索相比，速度至少提高了100倍，同时在许多大规模数据集上返回前10个MIPS，准确率至少为90%。
英文摘要：
We propose a novel dimensionality reduction method for maximum inner product search (MIPS), named CEOs, based on the theory of concomitants of extreme order statistics. Utilizing the asymptotic behavior of these concomitants, we show that a few dimensions associated with the extreme values of the query signature are enough to estimate inner products. Since CEOs only uses the sign of a small subset of the query signature for estimation, we can precompute all inner product estimators accurately before querying. These properties yield a sublinear MIPS algorithm with an exponential indexing space complexity. We show that our exponential space is optimal for the $(1 + ε)$-approximate MIPS on a unit sphere. The search recall of CEOs can be theoretically guaranteed under a mild condition. To deal with the exponential space complexity, we propose two practical variants, including sCEOs-TA and coCEOs, that use linear space for solving MIPS. sCEOs-TA exploits the threshold algorithm (TA) and provides superior search recalls to competitive MIPS solvers. coCEOs is a data and dimension co-reduction technique and outperforms sCEOs-TA on high recall requirements. Empirically, they are very simple to implement and achieve at least 100x speedup compared to the bruteforce search while returning top-10 MIPS with accuracy at least 90% on many large-scale data sets.
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