The rapid development of vector databases and large language models has significantly increased the importance of the Approximate Maximum Inner Product Search (AMIPS) problem. Over the past decade, various approaches have been proposed to efficiently address AMIPS problems. Compared to other methods, such as those based on Locality Sensitive Hashing (LSH), proximity graph-based methods have shown superior query performance on AMIPS. However, the performance of graph-based methods for AMIPS is still not fully optimized due to the norm bias issue, while graph-based methods have demonstrated strong effectiveness for ANNS. In this paper, we theoretically analyze norm bias in MIPS and identify a norm domination phenomenon: results are consistently dominated by a handful of large-norm vectors, with negligible contributions from small-norm ones. Building on this observation, we propose the Norm-Adaptive Partitioning (NAP) scheme, which splits the vectors in a dataset into H ead , B ody , and T ail based on vector norms. The H ead contains a small number of large-norm vectors for exhaustive search; the T ail includes small-norm vectors that can be safely pruned under accuracy bounds; and the B ody , characterized by concentrated norms, is well-suited for conventional proximity graphs. The main challenge of the NAP strategy is balancing query accuracy and the extra cost introduced by NAP by efficiently determining the sizes of ail , H ead , and B ody . To address this, we further design a practical NAP algorithm that minimizes search cost while ensuring bounded accuracy. To demonstrate NAP, we introduce a hybrid index combining a lightweight structure (for example, a hash table) for the head and a proximity graph for the body. Experimental results show that NAP reduces the index size of existing proximity graphs by over 50%, and the NAP-based hybrid index enables more than 2x speedup over state-of-the-art graph-based AMIPS methods at the same recall level.
Zhao et al. (Mon,) studied this question.