Understanding and Improving Proximity Graph based Maximum Inner Product Search

The inner-product navigable small world graph (ip-NSW) represents the state-of-the-art method for approximate maximum inner product search (MIPS) and it can achieve an order of magnitude speedup over the fastest baseline. However, to date it is still unclear where its exceptional performance comes from. In this paper, we show that there is a strong norm bias in the MIPS problem, which means that the large norm items are very likely to become the result of MIPS. Then we explain the good performance of ip-NSW as matching the norm bias of the MIPS problem - large norm items have big in-degrees in the ip-NSW proximity graph and a walk on the graph spends the majority of computation on these items, thus effectively avoids unnecessary computation on small norm items. Furthermore, we propose the ip-NSW+ algorithm, which improves ip-NSW by introducing an additional angular proximity graph. Search is first conducted on the angular graph to find the angular neighbors of a query and then the MIPS neighbors of these angular neighbors are used to initialize the candidate pool for search on the inner-product proximity graph. Experiment results show that ip-NSW+ consistently and significantly outperforms ip-NSW and provides more robust performance under different data distributions.