2-Cats: 2D Copula Approximating Transforms

TL;DR

This paper introduces 2-Cats, a neural network model that learns bivariate copulas directly without assuming specific families, ensuring theoretical properties and improved performance on diverse datasets.

Contribution

The paper presents a novel neural network approach for learning 2D copulas that does not depend on predefined copula families and incorporates derivative learning for better modeling.

Findings

Outperforms state-of-the-art methods on multiple datasets

Ensures theoretical copula properties in learned models

Extends training to include derivatives of the copula

Abstract

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 $C: (u,v)\in \mathbb{I}^2 \rightarrow \mathbb{I}$, where $\mathbb{I} = [0, 1]$. 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.