When Marginals Match but Structure Fails: Covariance Fidelity in Generative Models

TL;DR

This paper introduces a covariance-level dependence fidelity measure to evaluate whether generative models preserve the joint structure of data, addressing limitations of marginal distribution matching.

Contribution

The authors propose a new criterion, D_Sigma, for assessing dependence fidelity in generative models, and demonstrate its effectiveness across multiple data domains.

Findings

D_Sigma can be large even when marginals match perfectly.

Covariance divergence can cause instability in downstream tasks.

Bounding D_Sigma ensures stability in dependence-sensitive procedures.

Abstract

Generative models are increasingly deployed as substitutes for real data in downstream scientific workflows, yet standard evaluation criteria remain focused on marginal distribution matching. We argue that this represents a fundamental gap: downstream inference is rarely a marginal operation, and a model that passes every univariate diagnostic can still produce structurally unreliable synthetic data. We introduce covariance-level dependence fidelity, measured by D_Sigma(P,Q) = ||Sigma_P - Sigma_Q||_F, as a principled, computable criterion for evaluating whether a generative model preserves the joint structure of data beyond its univariate marginals. Three results formalise this criterion. First, marginal fidelity provides no constraint on dependence structure: D_Sigma can be made arbitrarily large while all univariate marginals match exactly. Second, covariance divergence induces quantifiable downstream instability, including sign reversals in population regression coefficients. Third, bounding D_Sigma provides positive stability guarantees for dependence-sensitive procedures such as PCA via Davis-Kahan-type bounds. Empirical validation across three domains, image data (Fashion-MNIST VAE, n = 60,000), bulk RNA-seq (TCGA-BRCA, n = 1,111), and a small-sample stress test (Alzheimer's gene expression, n = 113), shows that D_Sigma/delta consistently distinguishes structure-discarding from structure-preserving generators in cases where standard marginal diagnostics show little separation, confirming that covariance-level fidelity provides information orthogonal to existing evaluation metrics across domains and sample sizes.