# Invertible Generative Modeling using Linear Rational Splines

arXiv: https://arxiv.org/abs/2001.05168 · HTML: https://mlsanity.com/paper/2001.05168/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jan 2020.

## Built on by (most cited first, 30 of 40)

- [Learning Temporally Causal Latent Processes from General Temporal Data](https://mlsanity.com/paper/2110.05428/) (2021)
- [Learning Disentangled Representations in the Imaging Domain](https://mlsanity.com/paper/2108.12043/) (2021)
- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [Rhino: Deep Causal Temporal Relationship Learning With History-dependent Noise](https://mlsanity.com/paper/2210.14706/) (2022)
- [HuManiFlow: Ancestor-Conditioned Normalising Flows on SO(3) Manifolds for Human Pose and Shape Distribution Estimation](https://mlsanity.com/paper/2305.06968/) (2023)
- [A Structural Causal Model for MR Images of Multiple Sclerosis](https://mlsanity.com/paper/2103.03158/) (2021)
- [Normalizing Flows for Interventional Density Estimation](https://mlsanity.com/paper/2209.06203/) (2022)
- [Adversarial robustness of amortized Bayesian inference](https://mlsanity.com/paper/2305.14984/) (2023)
- [A Neural Framework for Generalized Causal Sensitivity Analysis](https://mlsanity.com/paper/2311.16026/) (2023)
- [Autoregressive Quantile Flows for Predictive Uncertainty Estimation](https://mlsanity.com/paper/2112.04643/) (2021)
- [Reliable Off-Policy Learning for Dosage Combinations](https://mlsanity.com/paper/2305.19742/) (2023)
- [Harmonization with Flow-based Causal Inference](https://mlsanity.com/paper/2106.06845/) (2021)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Counterfactual Generation with Identifiability Guarantees](https://mlsanity.com/paper/2402.15309/) (2024)
- [Deep Structural Causal Shape Models](https://mlsanity.com/paper/2208.10950/) (2022)
- [Semi-Autoregressive Energy Flows: Exploring Likelihood-Free Training of Normalizing Flows](https://mlsanity.com/paper/2206.06672/) (2022)
- [On the Robustness of Normalizing Flows for Inverse Problems in Imaging](https://mlsanity.com/paper/2212.04319/) (2022)
- [Deep Generative Models, Synthetic Tabular Data, and Differential Privacy: An Overview and Synthesis](https://mlsanity.com/paper/2307.15424/) (2023)
- [Transformer Neural Autoregressive Flows](https://mlsanity.com/paper/2401.01855/) (2024)
- [Preconditioned Neural Posterior Estimation for Likelihood-free Inference](https://mlsanity.com/paper/2404.13557/) (2024)
- [Bootstrap Your Flow](https://mlsanity.com/paper/2111.11510/) (2021)
- [One-Line-of-Code Data Mollification Improves Optimization of Likelihood-based Generative Models](https://mlsanity.com/paper/2305.18900/) (2023)
- [One flow to correct them all: improving simulations in high-energy physics with a single normalising flow and a switch](https://mlsanity.com/paper/2403.18582/) (2024)
- [Efficient Multimodal Sampling via Tempered Distribution Flow](https://mlsanity.com/paper/2304.03933/) (2023)
- [LiRA: Light-Robust Adversary for Model-based Reinforcement Learning in Real World](https://mlsanity.com/paper/2409.19617/) (2024)
- [Neural Diffeomorphic Non-uniform B-spline Flows](https://mlsanity.com/paper/2304.04555/) (2023)
- [Estimation of Bivariate Structural Causal Models by Variational Gaussian Process Regression Under Likelihoods Parametrised by Normalising Flows](https://mlsanity.com/paper/2109.02521/) (2021)
- [A Novel Application of Conditional Normalizing Flows: Stellar Age Inference with Gyrochronology](https://mlsanity.com/paper/2307.08753/) (2023)
- [Variational Elliptical Processes](https://mlsanity.com/paper/2311.12566/) (2023)
- [Design of Restricted Normalizing Flow towards Arbitrary Stochastic Policy with Computational Efficiency](https://mlsanity.com/paper/2412.12894/) (2024)

## Abstract (arXiv)

Normalizing flows attempt to model an arbitrary probability distribution through a set of invertible mappings. These transformations are required to achieve a tractable Jacobian determinant that can be used in high-dimensional scenarios. The first normalizing flow designs used coupling layer mappings built upon affine transformations. The significant advantage of such models is their easy-to-compute inverse. Nevertheless, making use of affine transformations may limit the expressiveness of such models. Recently, invertible piecewise polynomial functions as a replacement for affine transformations have attracted attention. However, these methods require solving a polynomial equation to calculate their inverse. In this paper, we explore using linear rational splines as a replacement for affine transformations used in coupling layers. Besides having a straightforward inverse, inference and generation have similar cost and architecture in this method. Moreover, simulation results demonstrate the competitiveness of this approach's performance compared to existing methods.