# Stacked Quantizers for Compositional Vector Compression

arXiv: https://arxiv.org/abs/1411.2173 · HTML: https://mlsanity.com/paper/1411.2173/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2014.

## Built on by (most cited first, 30 of 30)

- [Autoregressive Image Generation using Residual Quantization](https://mlsanity.com/paper/2203.01941/) (2022)
- [MoMask: Generative Masked Modeling of 3D Human Motions](https://mlsanity.com/paper/2312.00063/) (2023)
- [MeshGPT: Generating Triangle Meshes with Decoder-Only Transformers](https://mlsanity.com/paper/2311.15475/) (2023)
- [From Matching to Generation: A Survey on Generative Information Retrieval](https://mlsanity.com/paper/2404.14851/) (2024)
- [PivotMesh: Generic 3D Mesh Generation via Pivot Vertices Guidance](https://mlsanity.com/paper/2405.16890/) (2024)
- [MoGenTS: Motion Generation based on Spatial-Temporal Joint Modeling](https://mlsanity.com/paper/2409.17686/) (2024)
- [MeshArt: Generating Articulated Meshes with Structure-Guided Transformers](https://mlsanity.com/paper/2412.11596/) (2024)
- [A Survey of Multimodal Retrieval-Augmented Generation](https://mlsanity.com/paper/2504.08748/) (2025)
- [Multiplying Matrices Without Multiplying](https://mlsanity.com/paper/2106.10860/) (2021)
- [Not All Image Regions Matter: Masked Vector Quantization for Autoregressive Image Generation](https://mlsanity.com/paper/2305.13607/) (2023)
- [StrokeNUWA: Tokenizing Strokes for Vector Graphic Synthesis](https://mlsanity.com/paper/2401.17093/) (2024)
- [FoldToken: Learning Protein Language via Vector Quantization and Beyond](https://mlsanity.com/paper/2403.09673/) (2024)
- [IRGen: Generative Modeling for Image Retrieval](https://mlsanity.com/paper/2303.10126/) (2023)
- [UniCode: Learning a Unified Codebook for Multimodal Large Language Models](https://mlsanity.com/paper/2403.09072/) (2024)
- [BrepGPT: Autoregressive B-rep Generation with Voronoi Half-Patch](https://mlsanity.com/paper/2511.22171/) (2025)
- [Node Identifiers: Compact, Discrete Representations for Efficient Graph Learning](https://mlsanity.com/paper/2405.16435/) (2024)
- [MotionDreamer: One-to-Many Motion Synthesis with Localized Generative Masked Transformer](https://mlsanity.com/paper/2504.08959/) (2025)
- [Textual Decomposition Then Sub-motion-space Scattering for Open-Vocabulary Motion Generation](https://mlsanity.com/paper/2411.04079/) (2024)
- [Discrete Tokenization for Multimodal LLMs: A Comprehensive Survey](https://mlsanity.com/paper/2507.22920/) (2025)
- [VQ-Flow: Taming Normalizing Flows for Multi-Class Anomaly Detection via Hierarchical Vector Quantization](https://mlsanity.com/paper/2409.00942/) (2024)
- [MeToken: Uniform Micro-environment Token Boosts Post-Translational Modification Prediction](https://mlsanity.com/paper/2411.01856/) (2024)
- [MimicParts: Part-aware Style Injection for Speech-Driven 3D Motion Generation](https://mlsanity.com/paper/2510.13208/) (2025)
- [BoxSplitGen: A Generative Model for 3D Part Bounding Boxes in Varying Granularity](https://mlsanity.com/paper/2602.20666/) (2026)
- [Asymmetric Generative Recommendation via Kronecker Residual Bridge and Multi-Faceted Hierarchical Quantization](https://mlsanity.com/paper/2605.14512/) (2026)
- [Next-Token Prediction Learns Generalisable Representations of Sleep Physiology](https://mlsanity.com/paper/2606.09605/) (2026)
- [Image-Conditioned 3D Gaussian Splat Quantization](https://mlsanity.com/paper/2508.15372/) (2025)
- [GemiRec: Interest Quantization and Generation for Multi-Interest Recommendation](https://mlsanity.com/paper/2510.14626/) (2025)
- [LooC: Effective Low-Dimensional Codebook for Compositional Vector Quantization](https://mlsanity.com/paper/2601.00222/) (2026)
- [GSRQ: Gain-Shape Residual Quantization for Sub-1-bit KV Cache](https://mlsanity.com/paper/2607.01065/) (2026)
- [OrBIT: Structure-Guided Embedding Compression](https://mlsanity.com/paper/2610.10385/) (2026)

## Abstract (arXiv)

Recently, Babenko and Lempitsky introduced Additive Quantization (AQ), a generalization of Product Quantization (PQ) where a non-independent set of codebooks is used to compress vectors into small binary codes. Unfortunately, under this scheme encoding cannot be done independently in each codebook, and optimal encoding is an NP-hard problem. In this paper, we observe that PQ and AQ are both compositional quantizers that lie on the extremes of the codebook dependence-independence assumption, and explore an intermediate approach that exploits a hierarchical structure in the codebooks. This results in a method that achieves quantization error on par with or lower than AQ, while being several orders of magnitude faster. We perform a complexity analysis of PQ, AQ and our method, and evaluate our approach on standard benchmarks of SIFT and GIST descriptors, as well as on new datasets of features obtained from state-of-the-art convolutional neural networks.