# 2-Cats: 2D Copula Approximating Transforms

arXiv: https://arxiv.org/abs/2309.16391 · HTML: https://mlsanity.com/paper/2309.16391/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 73 references that are in the map.
- First posted on arXiv 28 Sep 2023.

## Builds on (most cited first, 22 of 22)

- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [Achieve the Minimum Width of Neural Networks for Universal Approximation](https://mlsanity.com/paper/2209.11395/) (2022)
- [Neural Likelihoods via Cumulative Distribution Functions](https://mlsanity.com/paper/1811.00974/) (2018)
- [Deep Archimedean Copulas](https://mlsanity.com/paper/2012.03137/) (2020)
- [PSD Representations for Effective Probability Models](https://mlsanity.com/paper/2106.16116/) (2021)
- [Implicit Generative Copulas](https://mlsanity.com/paper/2109.14567/) (2021)
- [Nonparametric estimation of simplified vine copula models: comparison of methods](https://mlsanity.com/paper/1701.00845/) (2017)
- [Squared Neural Families: A New Class of Tractable Density Models](https://mlsanity.com/paper/2305.13552/) (2023)
- [Probit transformation for nonparametric kernel estimation of the copula density](https://mlsanity.com/paper/1404.4414/) (2014)
- [Generative Archimedean Copulas](https://mlsanity.com/paper/2102.11351/) (2021)
- [Retrospective Uncertainties for Deep Models using Vine Copulas](https://mlsanity.com/paper/2302.12606/) (2023)
- [Copula-like Variational Inference](https://mlsanity.com/paper/1904.07153/) (2019)
- [Nonparametric estimation of copulas and copula densities by orthogonal projections](https://mlsanity.com/paper/2010.15351/) (2020)
- [Nonparametric Universal Copula Modeling](https://mlsanity.com/paper/1912.05503/) (2019)
- [Fixed Integral Neural Networks](https://mlsanity.com/paper/2307.14439/) (2023)

## Abstract (arXiv)

Copulas are powerful statistical tools for capturing dependencies across data dimensions. Applying Copulas involves estimating independent marginals, a straightforward task, followed by the much more challenging task of determining a single copulating function, C, that links these marginals. For bivariate data, a copula takes the form of a two-increasing function C: (u,v)∈ mathbbI2 rightarrow mathbbI, where mathbbI = [0, 1]. This paper proposes 2-Cats, a Neural Network (NN) model that learns two-dimensional Copulas without relying on specific Copula families (e.g., Archimedean). Furthermore, via both theoretical properties of the model and a Lagrangian training approach, we show that 2-Cats meets the desiderata of Copula properties. Moreover, inspired by the literature on Physics-Informed Neural Networks and Sobolev Training, we further extend our training strategy to learn not only the output of a Copula but also its derivatives. Our proposed method exhibits superior performance compared to the state-of-the-art across various datasets while respecting (provably for most and approximately for a single other) properties of C.