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Generative Correlation Manifolds: Generating Synthetic Data with Preserved Higher-Order Correlations

Jens E. d'Hondt, Wieger R. Punter, Odysseas Papapetrou

TL;DR

This work tackles privacy-preserving synthetic data by targeting the complete correlation structure of a dataset. It introduces Generative Correlation Manifolds (GCM), which uses the Cholesky decomposition of the target pairwise correlation matrix $C_{D,2}$ to map standard normal noise and then denormalizes to match the original feature means and variances. The central claim is that all $k$-th order correlations $C_{S,k}$ with $k \le m_S$ are determined by $C_{D,2}$, so preserving pairwise correlations guarantees preservation of higher-order structure; the method is non-iterative and operates in $O(n^3)$ time for the decomposition and $O(m_S n^2)$ for data generation, with open-source availability. This approach enables privacy-preserving data sharing, robust augmentation, and high-fidelity simulations in domains where complex feature interactions matter, while providing a clear theoretical framework for correlation preservation.

Abstract

The increasing need for data privacy and the demand for robust machine learning models have fueled the development of synthetic data generation techniques. However, current methods often succeed in replicating simple summary statistics but fail to preserve both the pairwise and higher-order correlation structure of the data that define the complex, multi-variable interactions inherent in real-world systems. This limitation can lead to synthetic data that is superficially realistic but fails when used for sophisticated modeling tasks. In this white paper, we introduce Generative Correlation Manifolds (GCM), a computationally efficient method for generating synthetic data. The technique uses Cholesky decomposition of a target correlation matrix to produce datasets that, by mathematical proof, preserve the entire correlation structure -- from simple pairwise relationships to higher-order interactions -- of the source dataset. We argue that this method provides a new approach to synthetic data generation with potential applications in privacy-preserving data sharing, robust model training, and simulation.

Generative Correlation Manifolds: Generating Synthetic Data with Preserved Higher-Order Correlations

TL;DR

This work tackles privacy-preserving synthetic data by targeting the complete correlation structure of a dataset. It introduces Generative Correlation Manifolds (GCM), which uses the Cholesky decomposition of the target pairwise correlation matrix to map standard normal noise and then denormalizes to match the original feature means and variances. The central claim is that all -th order correlations with are determined by , so preserving pairwise correlations guarantees preservation of higher-order structure; the method is non-iterative and operates in time for the decomposition and for data generation, with open-source availability. This approach enables privacy-preserving data sharing, robust augmentation, and high-fidelity simulations in domains where complex feature interactions matter, while providing a clear theoretical framework for correlation preservation.

Abstract

The increasing need for data privacy and the demand for robust machine learning models have fueled the development of synthetic data generation techniques. However, current methods often succeed in replicating simple summary statistics but fail to preserve both the pairwise and higher-order correlation structure of the data that define the complex, multi-variable interactions inherent in real-world systems. This limitation can lead to synthetic data that is superficially realistic but fails when used for sophisticated modeling tasks. In this white paper, we introduce Generative Correlation Manifolds (GCM), a computationally efficient method for generating synthetic data. The technique uses Cholesky decomposition of a target correlation matrix to produce datasets that, by mathematical proof, preserve the entire correlation structure -- from simple pairwise relationships to higher-order interactions -- of the source dataset. We argue that this method provides a new approach to synthetic data generation with potential applications in privacy-preserving data sharing, robust model training, and simulation.
Paper Structure (24 sections, 2 theorems, 7 equations, 1 figure)

This paper contains 24 sections, 2 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Deep Learning-Based Methods.
  4. Statistical Approaches.
  5. Our Contribution
  6. The Challenge of Capturing Higher-Order Correlations
  7. Methodology: The GCM Method
  8. Problem Formulation
  9. Objective
  10. Intuition
  11. Theoretical Foundation
  12. Process
  13. Computational Complexity
  14. Use Cases and Applications
  15. Call for Collaboration
  16. ...and 9 more sections

Key Result

Theorem 1

Let $D \in \mathbb{R}^{m_D \times n}$ be a dataset with column-wise means $\mu_D \in \mathbb{R}^n$, standard deviations $\sigma_D \in \mathbb{R}^n$, and correlation matrix $C_{D,2}$. A synthetic dataset $S$ with $n$ features and $m_S$ dimensions generated through Cholesky decomposition of $C_{D,2}$,

Figures (1)

  • Figure 1: Illustration of the different orders of correlation.

Theorems & Definitions (2)

  • Theorem 1
  • Lemma 1: K-order Correlation as a Function of Pairwise Correlations