# Block Neural Autoregressive Flow

arXiv: https://arxiv.org/abs/1904.04676 · HTML: https://mlsanity.com/paper/1904.04676/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2019.

## Built on by (most cited first, 30 of 62)

- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [A Comprehensive Survey of AI-Generated Content (AIGC): A History of Generative AI from GAN to ChatGPT](https://mlsanity.com/paper/2303.04226/) (2023)
- [Concrete Score Matching: Generalized Score Matching for Discrete Data](https://mlsanity.com/paper/2211.00802/) (2022)
- [Normalizing Flows are Capable Generative Models](https://mlsanity.com/paper/2412.06329/) (2024)
- [Membership Inference Attacks against Synthetic Data through Overfitting Detection](https://mlsanity.com/paper/2302.12580/) (2023)
- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [Robust Neural Posterior Estimation and Statistical Model Criticism](https://mlsanity.com/paper/2210.06564/) (2022)
- [Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint](https://mlsanity.com/paper/2109.11375/) (2021)
- [Tensor-Train Density Estimation](https://mlsanity.com/paper/2108.00089/) (2021)
- [Universal approximation property of invertible neural networks](https://mlsanity.com/paper/2204.07415/) (2022)
- [Implicit Normalizing Flows](https://mlsanity.com/paper/2103.09527/) (2021)
- [Generalized Normalizing Flows via Markov Chains](https://mlsanity.com/paper/2111.12506/) (2021)
- [On minimax density estimation via measure transport](https://mlsanity.com/paper/2207.10231/) (2022)
- [Invertible DenseNets with Concatenated LipSwish](https://mlsanity.com/paper/2102.02694/) (2021)
- [Autoregressive Quantile Flows for Predictive Uncertainty Estimation](https://mlsanity.com/paper/2112.04643/) (2021)
- [Surrogate Likelihoods for Variational Annealed Importance Sampling](https://mlsanity.com/paper/2112.12194/) (2021)
- [Data Plagiarism Index: Characterizing the Privacy Risk of Data-Copying in Tabular Generative Models](https://mlsanity.com/paper/2406.13012/) (2024)
- [Density estimation on low-dimensional manifolds: an inflation-deflation approach](https://mlsanity.com/paper/2105.12152/) (2021)
- [Privacy Vulnerabilities in Marginals-based Synthetic Data](https://mlsanity.com/paper/2410.05506/) (2024)
- [Monotone, Bi-Lipschitz, and Polyak-Lojasiewicz Networks](https://mlsanity.com/paper/2402.01344/) (2024)
- [Simulation-based Inference for High-dimensional Data using Surjective Sequential Neural Likelihood Estimation](https://mlsanity.com/paper/2308.01054/) (2023)
- [Flexible Language Modeling in Continuous Space with Transformer-based Autoregressive Flows](https://mlsanity.com/paper/2507.00425/) (2025)
- [Expressive Monotonic Neural Networks](https://mlsanity.com/paper/2307.07512/) (2023)
- [Transformer Neural Autoregressive Flows](https://mlsanity.com/paper/2401.01855/) (2024)
- [Efficient and Generalized end-to-end Autonomous Driving System with Latent Deep Reinforcement Learning and Demonstrations](https://mlsanity.com/paper/2401.11792/) (2024)
- [Importance Corrected Neural JKO Sampling](https://mlsanity.com/paper/2407.20444/) (2024)
- [LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport](https://mlsanity.com/paper/2411.12726/) (2024)
- [Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows](https://mlsanity.com/paper/2209.10873/) (2022)
- [Bootstrap Your Flow](https://mlsanity.com/paper/2111.11510/) (2021)
- [AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows](https://mlsanity.com/paper/2206.02102/) (2022)

## Abstract (arXiv)

Normalising flows (NFS) map two density functions via a differentiable bijection whose Jacobian determinant can be computed efficiently. Recently, as an alternative to hand-crafted bijections, Huang et al. (2018) proposed neural autoregressive flow (NAF) which is a universal approximator for density functions. Their flow is a neural network (NN) whose parameters are predicted by another NN. The latter grows quadratically with the size of the former and thus an efficient technique for parametrization is needed. We propose block neural autoregressive flow (B-NAF), a much more compact universal approximator of density functions, where we model a bijection directly using a single feed-forward network. Invertibility is ensured by carefully designing each affine transformation with block matrices that make the flow autoregressive and (strictly) monotone. We compare B-NAF to NAF and other established flows on density estimation and approximate inference for latent variable models. Our proposed flow is competitive across datasets while using orders of magnitude fewer parameters.