2-Cats: 2D Copula Approximating Transforms

Copulas are powerful statistical tools for capturing dependencies across multiple 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 $\mathbb{I} = [0, 1]$. In this paper, we propose 2-Cats, a Neural Network (NN) model that learns two-dimensional Copulas while preserving their key properties, without relying on specific Copula families (e.g., Archimedean). Furthermore, we introduce a training strategy inspired by the literature on Physics-Informed Neural Networks and Sobolev Training. Our proposed method exhibits superior performance compared to the state-of-the-art across various datasets while maintaining the fundamental mathematical properties of a Copula.