# iVPF: Numerical Invertible Volume Preserving Flow for Efficient Lossless Compression

arXiv: https://arxiv.org/abs/2103.16211 · HTML: https://mlsanity.com/paper/2103.16211/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 42 references that are in the map.
- First posted on arXiv 30 Mar 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Searching for MobileNetV3](https://mlsanity.com/paper/1905.02244/) (2019)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [WaveGlow: A Flow-based Generative Network for Speech Synthesis](https://mlsanity.com/paper/1811.00002/) (2018)
- [Video Pixel Networks](https://mlsanity.com/paper/1610.00527/) (2016)
- [Practical Full Resolution Learned Lossless Image Compression](https://mlsanity.com/paper/1811.12817/) (2018)
- [Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)](https://mlsanity.com/paper/2001.04872/) (2020)
- [Practical Lossless Compression with Latent Variables using Bits Back Coding](https://mlsanity.com/paper/1901.04866/) (2019)
- [Integer Discrete Flows and Lossless Compression](https://mlsanity.com/paper/1905.07376/) (2019)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)
- [Bit-Swap: Recursive Bits-Back Coding for Lossless Compression with Hierarchical Latent Variables](https://mlsanity.com/paper/1905.06845/) (2019)
- [HiLLoC: Lossless Image Compression with Hierarchical Latent Variable Models](https://mlsanity.com/paper/1912.09953/) (2019)
- [IDF++: Analyzing and Improving Integer Discrete Flows for Lossless Compression](https://mlsanity.com/paper/2006.12459/) (2020)
- [Compression with Flows via Local Bits-Back Coding](https://mlsanity.com/paper/1905.08500/) (2019)
- [Learning Better Lossless Compression Using Lossy Compression](https://mlsanity.com/paper/2003.10184/) (2020)
- [Lossless Image Compression through Super-Resolution](https://mlsanity.com/paper/2004.02872/) (2020)
- [Learning Discrete Distributions by Dequantization](https://mlsanity.com/paper/2001.11235/) (2020)

## Built on by (most cited first, 27 of 27)

- [On the Identifiability of Nonlinear ICA: Sparsity and Beyond](https://mlsanity.com/paper/2206.07751/) (2022)
- [iFlow: Numerically Invertible Flows for Efficient Lossless Compression via a Uniform Coder](https://mlsanity.com/paper/2111.00965/) (2021)
- [Deep Lossy Plus Residual Coding for Lossless and Near-lossless Image Compression](https://mlsanity.com/paper/2209.04847/) (2022)
- [Lossless Compression with Probabilistic Circuits](https://mlsanity.com/paper/2111.11632/) (2021)
- [Large Language Model for Lossless Image Compression with Visual Prompts](https://mlsanity.com/paper/2502.16163/) (2025)
- [aiWave: Volumetric Image Compression with 3-D Trained Affine Wavelet-like Transform](https://mlsanity.com/paper/2203.05822/) (2022)
- [Large Language Models for Lossless Image Compression: Next-Pixel Prediction in Language Space is All You Need](https://mlsanity.com/paper/2411.12448/) (2024)
- [Embracing the chaos: analysis and diagnosis of numerical instability in variational flows](https://mlsanity.com/paper/2307.06957/) (2023)
- [Split Hierarchical Variational Compression](https://mlsanity.com/paper/2204.02071/) (2022)
- [PILC: Practical Image Lossless Compression with an End-to-end GPU Oriented Neural Framework](https://mlsanity.com/paper/2206.05279/) (2022)
- [Approximation capabilities of measure-preserving neural networks](https://mlsanity.com/paper/2106.10911/) (2021)
- [Practical Learned Lossless JPEG Recompression with Multi-Level Cross-Channel Entropy Model in the DCT Domain](https://mlsanity.com/paper/2203.16357/) (2022)
- [Fast Lossless Neural Compression with Integer-Only Discrete Flows](https://mlsanity.com/paper/2206.08869/) (2022)
- [Mixed Variational Flows for Discrete Variables](https://mlsanity.com/paper/2308.15613/) (2023)
- [Wavelet-Like Transform-Based Technology in Response to the Call for Proposals on Neural Network-Based Image Coding](https://mlsanity.com/paper/2403.05937/) (2024)
- [DualComp: End-to-End Learning of a Unified Dual-Modality Lossless Compressor](https://mlsanity.com/paper/2505.16256/) (2025)
- [LVPNet: A Latent-variable-based Prediction-driven End-to-end Framework for Lossless Compression of Medical Images](https://mlsanity.com/paper/2506.17983/) (2025)
- [OmniZip: Learning a Unified and Lightweight Lossless Compressor for Multi-Modal Data](https://mlsanity.com/paper/2602.22286/) (2026)
- [SEEC: Segmentation-Assisted Multi-Entropy Models for Learned Lossless Image Compression](https://mlsanity.com/paper/2509.07704/) (2025)
- [OSOA: One-Shot Online Adaptation of Deep Generative Models for Lossless Compression](https://mlsanity.com/paper/2111.01662/) (2021)
- [Learned Lossless JPEG Transcoding via Joint Lossy and Residual Compression](https://mlsanity.com/paper/2208.11673/) (2022)
- [Multi-Scale Architectures Matter: On the Adversarial Robustness of Flow-based Lossless Compression](https://mlsanity.com/paper/2208.12716/) (2022)
- [Neural Invertible Variable-degree Optical Aberrations Correction](https://mlsanity.com/paper/2304.05564/) (2023)
- [FLLIC: Functionally Lossless Image Compression](https://mlsanity.com/paper/2401.13616/) (2024)
- [Deep Robust Reversible Watermarking](https://mlsanity.com/paper/2503.02490/) (2025)
- [RAWIC: Bit-Depth Adaptive Lossless Raw Image Compression](https://mlsanity.com/paper/2603.28105/) (2026)
- [LUMI: Tokenizer-Agnostic LLM-Based Lossless Image Compression](https://mlsanity.com/paper/2607.08221/) (2026)

## Abstract (arXiv)

It is nontrivial to store rapidly growing big data nowadays, which demands high-performance lossless compression techniques. Likelihood-based generative models have witnessed their success on lossless compression, where flow based models are desirable in allowing exact data likelihood optimisation with bijective mappings. However, common continuous flows are in contradiction with the discreteness of coding schemes, which requires either 1) imposing strict constraints on flow models that degrades the performance or 2) coding numerous bijective mapping errors which reduces the efficiency. In this paper, we investigate volume preserving flows for lossless compression and show that a bijective mapping without error is possible. We propose Numerical Invertible Volume Preserving Flow (iVPF) which is derived from the general volume preserving flows. By introducing novel computation algorithms on flow models, an exact bijective mapping is achieved without any numerical error. We also propose a lossless compression algorithm based on iVPF. Experiments on various datasets show that the algorithm based on iVPF achieves state-of-the-art compression ratio over lightweight compression algorithms.