# Soft Convex Quantization: Revisiting Vector Quantization with Convex Optimization

arXiv: https://arxiv.org/abs/2310.03004 · HTML: https://mlsanity.com/paper/2310.03004/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 30 references that are in the map.
- First posted on arXiv 4 Oct 2023.

## Builds on (most cited first, 24 of 24)

- [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)
- [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)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (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)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Vector Quantized Diffusion Model for Text-to-Image Synthesis](https://mlsanity.com/paper/2111.14822/) (2021)
- [SoundStream: An End-to-End Neural Audio Codec](https://mlsanity.com/paper/2107.03312/) (2021)
- [Autoregressive Image Generation using Residual Quantization](https://mlsanity.com/paper/2203.01941/) (2022)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Jukebox: A Generative Model for Music](https://mlsanity.com/paper/2005.00341/) (2020)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Understanding Straight-Through Estimator in Training Activation Quantized Neural Nets](https://mlsanity.com/paper/1903.05662/) (2019)
- [PixelSNAIL: An Improved Autoregressive Generative Model](https://mlsanity.com/paper/1712.09763/) (2017)
- [Fast Decoding in Sequence Models using Discrete Latent Variables](https://mlsanity.com/paper/1803.03382/) (2018)
- [Straightening Out the Straight-Through Estimator: Overcoming Optimization Challenges in Vector Quantized Networks](https://mlsanity.com/paper/2305.08842/) (2023)
- [SQ-VAE: Variational Bayes on Discrete Representation with Self-annealed Stochastic Quantization](https://mlsanity.com/paper/2205.07547/) (2022)

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

- [Restructuring Vector Quantization with the Rotation Trick](https://mlsanity.com/paper/2410.06424/) (2024)
- [Representation Collapsing Problems in Vector Quantization](https://mlsanity.com/paper/2411.16550/) (2024)
- [Discrete Tokenization for Multimodal LLMs: A Comprehensive Survey](https://mlsanity.com/paper/2507.22920/) (2025)
- [DiVeQ: Differentiable Vector Quantization Using the Reparameterization Trick](https://mlsanity.com/paper/2509.26469/) (2025)
- [Balance of Number of Embedding and their Dimensions in Vector Quantization](https://mlsanity.com/paper/2407.04939/) (2024)
- [Remote Inference over Dynamic Links via Adaptive Rate Deep Task-Oriented Vector Quantization](https://mlsanity.com/paper/2501.02521/) (2025)
- [Training-Free Vector Quantization via Gaussian VAEs](https://mlsanity.com/paper/2512.06609/) (2025)
- [Early Quantization Shrinks Codebook: A Simple Fix for Diversity-Preserving Tokenization](https://mlsanity.com/paper/2603.17052/) (2026)
- [Generalized Radius and Integrated Codebook Transforms for Differentiable Vector Quantization](https://mlsanity.com/paper/2602.01140/) (2026)
- [What Makes a Good Medical Image Tokenizer? Rethinking Reconstruction and Generation in Medical Image Tokenization](https://mlsanity.com/paper/2609.24691/) (2026)

## Abstract (arXiv)

Vector Quantization (VQ) is a well-known technique in deep learning for extracting informative discrete latent representations. VQ-embedded models have shown impressive results in a range of applications including image and speech generation. VQ operates as a parametric K-means algorithm that quantizes inputs using a single codebook vector in the forward pass. While powerful, this technique faces practical challenges including codebook collapse, non-differentiability and lossy compression. To mitigate the aforementioned issues, we propose Soft Convex Quantization (SCQ) as a direct substitute for VQ. SCQ works like a differentiable convex optimization (DCO) layer: in the forward pass, we solve for the optimal convex combination of codebook vectors that quantize the inputs. In the backward pass, we leverage differentiability through the optimality conditions of the forward solution. We then introduce a scalable relaxation of the SCQ optimization and demonstrate its efficacy on the CIFAR-10, GTSRB and LSUN datasets. We train powerful SCQ autoencoder models that significantly outperform matched VQ-based architectures, observing an order of magnitude better image reconstruction and codebook usage with comparable quantization runtime.