This paper investigates a new yet challenging problem called Reverse $k$-Maximum Inner Product Search (R$k$MIPS). Given a query (item) vector, a set of item vectors, and a set of user vectors, the problem of R$k$MIPS aims to find a set of user vectors whose inner products with the query vector are one of the $k$ largest among the query and item vectors. We propose the first subquadratic-time algorithm, i.e., Shifting-aware Asymmetric Hashing (SAH), to tackle the R$k$MIPS problem. To speed up the Maximum Inner Product Search (MIPS) on item vectors, we design a shifting-invariant asymmetric transformation and develop a novel sublinear-time Shifting-Aware Asymmetric Locality Sensitive Hashing (SA-ALSH) scheme. Furthermore, we devise a new blocking strategy based on the Cone-Tree to effectively prune user vectors (in a batch). We prove that SAH achieves a theoretical guarantee for solving the RMIPS problem. Experimental results on five real-world datasets show that SAH runs 4$\sim$8$\times$ faster than the state-of-the-art methods for R$k$MIPS while achieving F1-scores of over 90\%. The code is available at \url{https://github.com/HuangQiang/SAH}.

该论文研究了一种名为Reverse k-Maximum Inner Product Search (RkMIPS)的新问题，并提出了一种名为Shifting-aware Asymmetric Hashing (SAH)的子二次时间算法来解决该问题。该算法基于一种不变的不对称变换，设计了Shifting-Aware Asymmetric Locality Sensitive Hashing (SA-ALSH)方案以加速item vectors上的Maximum Inner Product Search (MIPS)，并使用基于Cone-Tree的新型阻塞策略来批量有效地修剪用户向量。在5个真实世界的数据集上进行实验，证明SAH比现有的RkMIPS方法快4到8倍，同时实现了超过90％的F1-score。

SAH: 可感知轮换的非对称哈希用于反向$k$最大内积搜索