# A Unified Rate-Distortion Perspective on Vector, Product, and Scalar Quantization

arXiv: https://arxiv.org/abs/2609.02107 · HTML: https://mlsanity.com/paper/2609.02107/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 39 of its 53 references that are in the map.
- First posted on arXiv 2 Sep 2026.

## Builds on (most cited first, 30 of 39)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Generating Diverse High-Fidelity Images with VQ-VAE-2](https://mlsanity.com/paper/1906.00446/) (2019)
- [Visual Autoregressive Modeling: Scalable Image Generation via Next-Scale Prediction](https://mlsanity.com/paper/2404.02905/) (2024)
- [Autoregressive Model Beats Diffusion: Llama for Scalable Image Generation](https://mlsanity.com/paper/2406.06525/) (2024)
- [Autoregressive Image Generation using Residual Quantization](https://mlsanity.com/paper/2203.01941/) (2022)
- [End-to-end Optimized Image Compression](https://mlsanity.com/paper/1611.01704/) (2016)
- [Vector-quantized Image Modeling with Improved VQGAN](https://mlsanity.com/paper/2110.04627/) (2021)
- [Language Model Beats Diffusion -- Tokenizer is Key to Visual Generation](https://mlsanity.com/paper/2310.05737/) (2023)
- [Jukebox: A Generative Model for Music](https://mlsanity.com/paper/2005.00341/) (2020)
- [Finite Scalar Quantization: VQ-VAE Made Simple](https://mlsanity.com/paper/2309.15505/) (2023)
- [Reconstruction vs. Generation: Taming Optimization Dilemma in Latent Diffusion Models](https://mlsanity.com/paper/2501.01423/) (2025)
- [Infinity: Scaling Bitwise AutoRegressive Modeling for High-Resolution Image Synthesis](https://mlsanity.com/paper/2412.04431/) (2024)
- [An Image is Worth 32 Tokens for Reconstruction and Generation](https://mlsanity.com/paper/2406.07550/) (2024)
- [UniTok: A Unified Tokenizer for Visual Generation and Understanding](https://mlsanity.com/paper/2502.20321/) (2025)
- [Image and Video Tokenization with Binary Spherical Quantization](https://mlsanity.com/paper/2406.07548/) (2024)
- [FlexTok: Resampling Images into 1D Token Sequences of Flexible Length](https://mlsanity.com/paper/2502.13967/) (2025)
- [ImageFolder: Autoregressive Image Generation with Folded Tokens](https://mlsanity.com/paper/2410.01756/) (2024)
- [Straightening Out the Straight-Through Estimator: Overcoming Optimization Challenges in Vector Quantized Networks](https://mlsanity.com/paper/2305.08842/) (2023)
- [Addressing Representation Collapse in Vector Quantized Models with One Linear Layer](https://mlsanity.com/paper/2411.02038/) (2024)
- [SQ-VAE: Variational Bayes on Discrete Representation with Self-annealed Stochastic Quantization](https://mlsanity.com/paper/2205.07547/) (2022)
- [Scaling the Codebook Size of VQGAN to 100,000 with a Utilization Rate of 99%](https://mlsanity.com/paper/2406.11837/) (2024)
- [Hierarchical Quantized Autoencoders](https://mlsanity.com/paper/2002.08111/) (2020)

## Abstract (arXiv)

Discrete visual tokenization, predominantly driven by vector, scalar, and product quantization, lacks a unified conceptual framework for understanding quantization tradeoffs. In this paper, we propose a unified rate--distortion perspective on modern discrete visual tokenization. By viewing quantization as lossy compression, we characterize the nominal fixed-length coding rate through token count and codebook size, and quantization error as the distortion. Within this framework, we resolve three central questions. First, we theoretically and empirically show that minimizing distortion, rather than maximizing codebook utilization, is the primary intrinsic objective for reconstruction fidelity, with a direct connection to the STE-induced gradient discrepancy. Second, we establish two critical fairness conditions for intrinsic quantization comparison: controlling latent feature statistics and enforcing identical coding rates. Third, under these conditions, we recover the VQ--PQ--SQ distortion hierarchy in modern visual tokenization and show empirically that modern VQ methods achieve the lowest distortion. This work provides a foundational rate--distortion reframing of modern discrete visual tokenization, resolves ambiguities in quantizer evaluation, and provides a controlled framework for isolating intrinsic quantization effectiveness under fixed-rate constraints.