Optimal Post-processing of Synthetic Data for Pearson Correlation Matching

Preserving correlation in synthetic data is of interest in several applications. Existing approaches mainly address correlation preservation during the generation procedure. Here, we instead consider it as a postprocessing problem. Given original and synthetic tabular data, we seek the smallest change to the synthetic dataset that restores the Pearson correlation matrix of the original data. Under suitable assumptions, we derive a unique explicit solution to the minimization problem. We also provide a bound on the size of the correction in terms of the initial correlation error. On the numerical side, across several datasets and generation methods, the proposed approach largely preserves marginal distributions, t-SNE geometry, and classification performance. The method is generator-independent and can be applied after synthesis without modifying the generation procedure.