# Norm-Explicit Quantization: Improving Vector Quantization for Maximum Inner Product Search

arXiv: https://arxiv.org/abs/1911.04654 · HTML: https://mlsanity.com/paper/1911.04654/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Nov 2019.

## Built on by (most cited first, 10 of 10)

- [Embedding in Recommender Systems: A Survey](https://mlsanity.com/paper/2310.18608/) (2023)
- [Flashlight: Scalable Link Prediction with Effective Decoders](https://mlsanity.com/paper/2209.10100/) (2022)
- [Faster Maximum Inner Product Search in High Dimensions](https://mlsanity.com/paper/2212.07551/) (2022)
- [Probabilistic Routing for Graph-Based Approximate Nearest Neighbor Search](https://mlsanity.com/paper/2402.11354/) (2024)
- [Probabilistic Kernel Function for Fast Angle Testing](https://mlsanity.com/paper/2505.20274/) (2025)
- [SAH: Shifting-aware Asymmetric Hashing for Reverse k-Maximum Inner Product Search](https://mlsanity.com/paper/2211.12751/) (2022)
- [Amortized Maximum Inner Product Search with Learned Support Functions](https://mlsanity.com/paper/2603.08001/) (2026)
- [Fuzzy Norm-Explicit Product Quantization for Recommender Systems](https://mlsanity.com/paper/2412.06069/) (2024)
- [RSLM: Training-Free Vector Quantization for Approximate Nearest Neighbor Search](https://mlsanity.com/paper/2608.30384/) (2026)
- [PQ-HSA: Reusing Product-Quantized Scores for Hybrid Sparse-Approximate Attention](https://mlsanity.com/paper/2609.33746/) (2026)

## Abstract (arXiv)

Vector quantization (VQ) techniques are widely used in similarity search for data compression, fast metric computation and etc. Originally designed for Euclidean distance, existing VQ techniques (e.g., PQ, AQ) explicitly or implicitly minimize the quantization error. In this paper, we present a new angle to analyze the quantization error, which decomposes the quantization error into norm error and direction error. We show that quantization errors in norm have much higher influence on inner products than quantization errors in direction, and small quantization error does not necessarily lead to good performance in maximum inner product search (MIPS). Based on this observation, we propose norm-explicit quantization (NEQ) --- a general paradigm that improves existing VQ techniques for MIPS. NEQ quantizes the norms of items in a dataset explicitly to reduce errors in norm, which is crucial for MIPS. For the direction vectors, NEQ can simply reuse an existing VQ technique to quantize them without modification. We conducted extensive experiments on a variety of datasets and parameter configurations. The experimental results show that NEQ improves the performance of various VQ techniques for MIPS, including PQ, OPQ, RQ and AQ.