Title: 2-Cats: 2D Copula Approximating Transforms

Abstract: 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.
note: Dear readers, a reviewer correctly captured a mistake in our proof of P2. Given that arXiv does not allow removals, we are keeping this version with this note on Arxiv with this note while we correct this issue.