Distributional Evaluation of Generative Models via Relative Density Ratio

We propose a functional evaluation metric for generative models based on the relative density ratio (RDR) designed to characterize distributional differences between real and generated samples. We show that the RDR as a functional summary of the goodness-of-fit for the generative model, possesses several desirable theoretical properties. It preserves $\phi$-divergence between two distributions, enables sample-level evaluation that facilitates downstream investigations of feature-specific distributional differences, and has a bounded range that affords clear interpretability and numerical stability. Functional estimation of the RDR is achieved efficiently through convex optimization on the variational form of $\phi$-divergence. We provide theoretical convergence rate guarantees for general estimators based on M-estimator theory, as well as the convergence rates of neural network-based estimators when the true ratio is in the anisotropic Besov space. We demonstrate the power of the proposed RDR-based evaluation through numerical experiments on MNIST, CelebA64, and the American Gut project microbiome data. We show that the estimated RDR not only allows for an effective comparison of the overall performance of competing generative models, but it can also offer a convenient means of revealing the nature of the underlying goodness-of-fit. This enables one to assess support overlap, coverage, and fidelity while pinpointing regions of the sample space where generators concentrate and revealing the features that drive the most salient distributional differences.