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2-Cats: 2D Copula Approximating Transforms

Flavio Figueiredo, José Geraldo Fernandes, Jackson Silva, Renato M. Assunção

TL;DR

The paper tackles learning flexible 2D copulas without relying on fixed parametric families by introducing 2-Cats, a neural network that constructs a Copula transform H_{}{{\theta}}(u,v) = G(z(t_v(u)), z(t_u(v))) using monotone, integrally-constructed transforms to satisfy Copula properties. It enforces Copula desiderata via a Lagrangian framework to approximate boundary conditions (H_{}(u,1) \approx u and H_{}(1,v) \approx v) and augments training with Sobolev losses that align not only the Copula output but also its first and second derivatives, enabling derivative-aware sampling and Vine-like constructions. The approach is validated on synthetic and real datasets, showing competitive or superior negative log-likelihoods compared to baselines, and ablation studies highlight the value of each loss component and the Lagrangian term. The work advances Copula modeling by combining flexible neural transforms with principled constraint enforcement and derivative-informed training, with potential extensions to higher dimensions via Pair Copula Constructions. Overall, 2-Cats demonstrates strong practical impact for modeling multivariate dependencies without restrictive Copula families.

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.

2-Cats: 2D Copula Approximating Transforms

TL;DR

The paper tackles learning flexible 2D copulas without relying on fixed parametric families by introducing 2-Cats, a neural network that constructs a Copula transform H_{}{{\theta}}(u,v) = G(z(t_v(u)), z(t_u(v))) using monotone, integrally-constructed transforms to satisfy Copula properties. It enforces Copula desiderata via a Lagrangian framework to approximate boundary conditions (H_{}(u,1) \approx u and H_{}(1,v) \approx v) and augments training with Sobolev losses that align not only the Copula output but also its first and second derivatives, enabling derivative-aware sampling and Vine-like constructions. The approach is validated on synthetic and real datasets, showing competitive or superior negative log-likelihoods compared to baselines, and ablation studies highlight the value of each loss component and the Lagrangian term. The work advances Copula modeling by combining flexible neural transforms with principled constraint enforcement and derivative-informed training, with potential extensions to higher dimensions via Pair Copula Constructions. Overall, 2-Cats demonstrates strong practical impact for modeling multivariate dependencies without restrictive Copula families.

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, , that links these marginals. For bivariate data, a copula takes the form of a two-increasing function , where . 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.
Paper Structure (20 sections, 21 equations, 1 figure, 6 tables)

This paper contains 20 sections, 21 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. On the Desiderata of a Copula and 2-Cats Properties
  4. Sobolev Losses for 2-Cats Models
  5. Experimental Results
  6. Empirical First Derivative Estimator
  7. 2-Cats on Datasets (Without Lagrangian Term)
  8. On the Lagrangian Term for P3
  9. Conclusions
  10. On Derivatives and Integrals of NNs
  11. An Example on why approximating derivatives and integrals fail
  12. On the Integral of Neural Networks
  13. On the Derivative of Underlying Process and its impact on Neural Networks
  14. Sampling from 2-Cats
  15. Ablation Study
  16. ...and 5 more sections

Figures (1)

  • Figure 1: Models vs Derivatives. Here, we compare sample data from the simple $y = sin(x)$ process. Data is sampled in three settings, a "full" dataset with a larger range (left), a noisy dataset (middle), and a limited view (right). NN models (lines) approximate the data points in every case. However, the derivatives/integrals of the model do not approximate function derivatives (last row).

Theorems & Definitions (6)

  • proof
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