# Neural Copula: A unified framework for estimating generic high-dimensional Copula functions

arXiv: https://arxiv.org/abs/2205.15031 · HTML: https://mlsanity.com/paper/2205.15031/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 28 references that are in the map.
- First posted on arXiv 23 May 2022.

## Builds on (most cited first, 3 of 3)

- [Neural Likelihoods via Cumulative Distribution Functions](https://mlsanity.com/paper/1811.00974/) (2018)
- [Deep Archimedean Copulas](https://mlsanity.com/paper/2012.03137/) (2020)
- [Efficient and feasible inference for high-dimensional normal copula regression models](https://mlsanity.com/paper/2203.04619/) (2022)

## Built on by (most cited first, 7 of 7)

- [Copula Density Neural Estimation](https://mlsanity.com/paper/2211.15353/) (2022)
- [Relevance meets Diversity: A User-Centric Framework for Knowledge Exploration through Recommendations](https://mlsanity.com/paper/2408.03772/) (2024)
- [Marginal Causal Flows for Validation and Inference](https://mlsanity.com/paper/2411.01295/) (2024)
- [NN-Copula-CD: A Copula-Guided Interpretable Neural Network for Change Detection in Heterogeneous Remote Sensing Images](https://mlsanity.com/paper/2303.17448/) (2023)
- [Deep Generative Quantile Bayes](https://mlsanity.com/paper/2410.08378/) (2024)
- [Cross-Modal Alignment via Variational Copula Modelling](https://mlsanity.com/paper/2511.03196/) (2025)
- [Capturing Multivariate Dependencies of EV Charging Events: From Parametric Copulas to Neural Density Estimation](https://mlsanity.com/paper/2603.29554/) (2026)

## Abstract (arXiv)

The Copula is widely used to describe the relationship between the marginal distribution and joint distribution of random variables. The estimation of high-dimensional Copula is difficult, and most existing solutions rely either on simplified assumptions or on complicating recursive decompositions. Therefore, people still hope to obtain a generic Copula estimation method with both universality and simplicity. To reach this goal, a novel neural network-based method (named Neural Copula) is proposed in this paper. In this method, a hierarchical unsupervised neural network is constructed to estimate the marginal distribution function and the Copula function by solving differential equations. In the training program, various constraints are imposed on both the neural network and its derivatives. The Copula estimated by the proposed method is smooth and has an analytic expression. The effectiveness of the proposed method is evaluated on both real-world datasets and complex numerical simulations. Experimental results show that Neural Copula's fitting quality for complex distributions is much better than classical methods. The relevant code for the experiments is available on GitHub. (We encourage the reader to run the program for a better understanding of the proposed method).