# Jacobian Determinant of Normalizing Flows

arXiv: https://arxiv.org/abs/2102.06539 · HTML: https://mlsanity.com/paper/2102.06539/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 39 references that are in the map.
- First posted on arXiv 12 Feb 2021.

## Builds on (most cited first, 30 of 33)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Squeeze-and-Excitation Networks](https://mlsanity.com/paper/1709.01507/) (2017)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [Sylvester Normalizing Flows for Variational Inference](https://mlsanity.com/paper/1803.05649/) (2018)
- [Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models](https://mlsanity.com/paper/2002.07101/) (2020)
- [Approximation Capabilities of Neural ODEs and Invertible Residual Networks](https://mlsanity.com/paper/1907.12998/) (2019)
- [The Expressive Power of a Class of Normalizing Flow Models](https://mlsanity.com/paper/2006.00392/) (2020)
- [Monge-Amp`ere Flow for Generative Modeling](https://mlsanity.com/paper/1809.10188/) (2018)
- [Potential Flow Generator with L2 Optimal Transport Regularity for Generative Models](https://mlsanity.com/paper/1908.11462/) (2019)

## Built on by (most cited first, 8 of 8)

- [Lagrangian Flow Networks for Conservation Laws](https://mlsanity.com/paper/2305.16846/) (2023)
- [Conditional Matrix Flows for Gaussian Graphical Models](https://mlsanity.com/paper/2306.07255/) (2023)
- [Distribution-aware Goal Prediction and Conformant Model-based Planning for Safe Autonomous Driving](https://mlsanity.com/paper/2212.08729/) (2022)
- [Injective flows for star-like manifolds](https://mlsanity.com/paper/2406.09116/) (2024)
- [VeriFlow: Modeling Distributions for Neural Network Verification](https://mlsanity.com/paper/2406.14265/) (2024)
- [PV Fleet Modeling via Smooth Periodic Gaussian Copula](https://mlsanity.com/paper/2307.00004/) (2023)
- [Marginal Flow: a flexible and efficient framework for density estimation](https://mlsanity.com/paper/2509.26221/) (2025)
- [On Uniformly Scaling Flows: A Density-Aligned Approach to Deep One-Class Classification](https://mlsanity.com/paper/2510.09452/) (2025)

## Abstract (arXiv)

Normalizing flows learn a diffeomorphic mapping between the target and base distribution, while the Jacobian determinant of that mapping forms another real-valued function. In this paper, we show that the Jacobian determinant mapping is unique for the given distributions, hence the likelihood objective of flows has a unique global optimum. In particular, the likelihood for a class of flows is explicitly expressed by the eigenvalues of the auto-correlation matrix of individual data point, and independent of the parameterization of neural network, which provides a theoretical optimal value of likelihood objective and relates to probabilistic PCA. Additionally, Jacobian determinant is a measure of local volume change and is maximized when MLE is used for optimization. To stabilize normalizing flows training, it is required to maintain a balance between the expansiveness and contraction of volume, meaning Lipschitz constraint on the diffeomorphic mapping and its inverse. With these theoretical results, several principles of designing normalizing flow were proposed. And numerical experiments on highdimensional datasets (such as CelebA-HQ 1024x1024) were conducted to show the improved stability of training.