Reverse Maximum Inner Product Search: Formulation, Algorithms, and Analysis

Abstract

The MIPS (maximum inner product search), which finds the item with the highest inner product with a given query user, is an essential problem in the recommendation field. Usually, e-commerce companies face situations where they want to promote and sell new or discounted items. In these situations, we have to consider a question: who are interested in the items and how to find them? This article answers this question by addressing a new problem called reverse maximum inner product search (reverse MIPS). Given a query vector and two sets of vectors (user vectors and item vectors), the problem of reverse MIPS finds a set of user vectors whose inner product with the query vector is the maximum among the query and item vectors. Although the importance of this problem is clear, its straightforward implementation incurs a computationally expensive cost. We therefore propose Simpfer, a simple, fast, and exact algorithm for reverse MIPS. In an offline phase, Simpfer builds a simple index that maintains a lower-bound of the maximum inner product. By exploiting this index, Simpfer judges whether the query vector can have the maximum inner product or not, for a given user vector, in a constant time. Our index enables filtering user vectors, which cannot have the maximum inner product with the query vector, in a batch. We theoretically demonstrate that Simpfer outperforms baselines employing state-of-the-art MIPS techniques. In addition, we answer two new research questions. Can approximation algorithms further improve reverse MIPS processing? Is there an exact algorithm that is faster than Simpfer? For the former, we show that approximation with quality guarantee provides a little speed-up. For the latter, we propose Simpfer++, a theoretically and practically faster algorithm than Simpfer. Our extensive experiments on real datasets show that Simpfer is at least two orders of magnitude faster than the baselines, and Simpfer++ further improves the online processing time.