On the Statistical Complexity of Estimating VENDI Scores from Empirical Data

Reference-free evaluation metrics for generative models have recently been studied in the machine learning community. As a reference-free metric, the VENDI score quantifies the diversity of generative models using matrix-based entropy from information theory. The VENDI score is usually computed through the eigendecomposition of an $n \times n$ kernel matrix for $n$ generated samples. However, due to the high computational cost of eigendecomposition for large $n$, the score is often computed on sample sizes limited to a few tens of thousands. In this paper, we explore the statistical convergence of the VENDI score and demonstrate that for kernel functions with an infinite feature map dimension, the evaluated score for a limited sample size may not converge to the matrix-based entropy statistic. We introduce an alternative statistic called the $t$-truncated VENDI statistic. We show that the existing Nystr\"om method and the FKEA approximation method for the VENDI score will both converge to the defined truncated VENDI statistic given a moderate sample size. We perform several numerical experiments to illustrate the concentration of the empirical VENDI score around the truncated VENDI statistic and discuss how this statistic correlates with the visual diversity of image data.