# Pushing Toward the Simplex Vertices: A Simple Remedy for Code Collapse in Smoothed Vector Quantization

arXiv: https://arxiv.org/abs/2509.22161 · HTML: https://mlsanity.com/paper/2509.22161/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 36 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [wav2vec 2.0: A Framework for Self-Supervised Learning of Speech Representations](https://mlsanity.com/paper/2006.11477/) (2020)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Scaling Autoregressive Models for Content-Rich Text-to-Image Generation](https://mlsanity.com/paper/2206.10789/) (2022)
- [Neural Codec Language Models are Zero-Shot Text to Speech Synthesizers](https://mlsanity.com/paper/2301.02111/) (2023)
- [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)
- [vq-wav2vec: Self-Supervised Learning of Discrete Speech Representations](https://mlsanity.com/paper/1910.05453/) (2019)
- [Speak Foreign Languages with Your Own Voice: Cross-Lingual Neural Codec Language Modeling](https://mlsanity.com/paper/2303.03926/) (2023)
- [Addressing Representation Collapse in Vector Quantized Models with One Linear Layer](https://mlsanity.com/paper/2411.02038/) (2024)
- [The challenge of realistic music generation: modelling raw audio at scale](https://mlsanity.com/paper/1806.10474/) (2018)
- [Restructuring Vector Quantization with the Rotation Trick](https://mlsanity.com/paper/2410.06424/) (2024)
- [Vector Quantized Contrastive Predictive Coding for Template-based Music Generation](https://mlsanity.com/paper/2004.10120/) (2020)
- [Fast MLE Computation for the Dirichlet Multinomial](https://mlsanity.com/paper/1405.0099/) (2014)

## Abstract (arXiv)

Vector quantization, which discretizes a continuous vector space into a finite set of representative vectors (a codebook), has been widely adopted in modern machine learning. Despite its effectiveness, vector quantization poses a fundamental challenge: the non-differentiable quantization step blocks gradient backpropagation. Smoothed vector quantization addresses this issue by relaxing the discrete selection of a codebook vector into a weighted combination of codebook entries, represented as the matrix product of a simplex vector and the codebook. Effective smoothing requires two properties: (1) smoothed code-selection vectors should remain close to a onehot vector, ensuring tight approximation, and (2) all codebook entries should be utilized, preventing code collapse. Existing methods typically address these desiderata separately. By contrast, the present study introduces a simple and intuitive regularization that promotes both simultaneously by minimizing the distance between each simplex vertex and its K-nearest smoothed code-selectors. Representative benchmarks on image encoding demonstrate that the proposed method achieves more effective codebook utilization and improves performance over prior approaches.