  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}.
