# IDF++: Analyzing and Improving Integer Discrete Flows for Lossless Compression

arXiv: https://arxiv.org/abs/2006.12459 · HTML: https://mlsanity.com/paper/2006.12459/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2020.

## Built on by (most cited first, 30 of 36)

- [Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions](https://mlsanity.com/paper/2102.05379/) (2021)
- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [Autoregressive Diffusion Models](https://mlsanity.com/paper/2110.02037/) (2021)
- [Unleashing Transformers: Parallel Token Prediction with Discrete Absorbing Diffusion for Fast High-Resolution Image Generation from Vector-Quantized Codes](https://mlsanity.com/paper/2111.12701/) (2021)
- [Soft then Hard: Rethinking the Quantization in Neural Image Compression](https://mlsanity.com/paper/2104.05168/) (2021)
- [On the Out-of-distribution Generalization of Probabilistic Image Modelling](https://mlsanity.com/paper/2109.02639/) (2021)
- [iVPF: Numerical Invertible Volume Preserving Flow for Efficient Lossless Compression](https://mlsanity.com/paper/2103.16211/) (2021)
- [iFlow: Numerically Invertible Flows for Efficient Lossless Compression via a Uniform Coder](https://mlsanity.com/paper/2111.00965/) (2021)
- [A Fast Transformer-based General-Purpose Lossless Compressor](https://mlsanity.com/paper/2203.16114/) (2022)
- [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)
- [Improving Lossless Compression Rates via Monte Carlo Bits-Back Coding](https://mlsanity.com/paper/2102.11086/) (2021)
- [Deep Generative Models for Detector Signature Simulation: A Taxonomic Review](https://mlsanity.com/paper/2312.09597/) (2023)
- [Semi-Discrete Normalizing Flows through Differentiable Tessellation](https://mlsanity.com/paper/2203.06832/) (2022)
- [Partition and Code: learning how to compress graphs](https://mlsanity.com/paper/2107.01952/) (2021)
- [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)
- [Lossless Compression with Latent Variable Models](https://mlsanity.com/paper/2104.10544/) (2021)
- [Parallel Neural Local Lossless Compression](https://mlsanity.com/paper/2201.05213/) (2022)
- [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)
- [Learning Non-linear Wavelet Transformation via Normalizing Flow](https://mlsanity.com/paper/2101.11306/) (2021)
- [WISE: A Framework for Gigapixel Whole-Slide-Image Lossless Compression](https://mlsanity.com/paper/2503.18074/) (2025)
- [Discrete Denoising Flows](https://mlsanity.com/paper/2107.11625/) (2021)
- [L3C-Stereo: Lossless Compression for Stereo Images](https://mlsanity.com/paper/2108.09422/) (2021)
- [Random Edge Coding: One-Shot Bits-Back Coding of Large Labeled Graphs](https://mlsanity.com/paper/2305.09705/) (2023)
- [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)
- [Survey on Visual Signal Coding and Processing with Generative Models: Technologies, Standards and Optimization](https://mlsanity.com/paper/2405.14221/) (2024)
- [SEEC: Segmentation-Assisted Multi-Entropy Models for Learned Lossless Image Compression](https://mlsanity.com/paper/2509.07704/) (2025)

## Abstract (arXiv)

In this paper we analyse and improve integer discrete flows for lossless compression. Integer discrete flows are a recently proposed class of models that learn invertible transformations for integer-valued random variables. Their discrete nature makes them particularly suitable for lossless compression with entropy coding schemes. We start by investigating a recent theoretical claim that states that invertible flows for discrete random variables are less flexible than their continuous counterparts. We demonstrate with a proof that this claim does not hold for integer discrete flows due to the embedding of data with finite support into the countably infinite integer lattice. Furthermore, we zoom in on the effect of gradient bias due to the straight-through estimator in integer discrete flows, and demonstrate that its influence is highly dependent on architecture choices and less prominent than previously thought. Finally, we show how different architecture modifications improve the performance of this model class for lossless compression, and that they also enable more efficient compression: a model with half the number of flow layers performs on par with or better than the original integer discrete flow model.