Evaluating the diversity of generative models without reference data poses methodological challenges. The reference-free Vendi and RKE scores address this by quantifying the diversity of generated data using matrix-based entropy measures. Among these two, the Vendi score is typically computed via the eigendecomposition of an $n \times n$ kernel matrix constructed from n generated samples. However, the prohibitive computational cost of eigendecomposition for large $n$ often limits the number of samples used to fewer than 20,000. In this paper, we investigate the statistical convergence of the Vendi and RKE scores under restricted sample sizes. We numerically demonstrate that, in general, the Vendi score computed with standard sample sizes below 20,000 may not converge to its asymptotic value under infinite sampling. To address this, we introduce the $t$-truncated Vendi score by truncating the eigenspectrum of the kernel matrix, which is provably guaranteed to converge to its population limit with $n=\mathcal{O}(t)$ samples. We further show that existing Nystr\"om and FKEA approximation methods converge to the asymptotic limit of the truncated Vendi score. In contrast to the Vendi score, we prove that the RKE score enjoys universal convergence guarantees across all kernel functions. We conduct several numerical experiments to illustrate the concentration of Nystr\"om and FKEA computed Vendi scores around the truncated Vendi score, and we analyze how the truncated Vendi and RKE scores correlate with the diversity of image and text data. The code is available at https://github.com/aziksh-ospanov/truncated-vendi.

翻译：在无参考数据条件下评估生成模型的多样性面临方法学挑战。无参考的Vendi分数和RKE分数通过基于矩阵的熵度量来量化生成数据的多样性，从而解决这一问题。其中，Vendi分数通常通过对由n个生成样本构建的$n \times n$核矩阵进行特征分解来计算。然而，对于较大的n，特征分解的计算成本过高，通常将使用的样本数量限制在20,000以下。本文研究了在有限样本量下Vendi分数和RKE分数的统计收敛性。我们通过数值实验证明，在标准样本量低于20,000的情况下，Vendi分数通常无法收敛到其在无限采样下的渐近值。为解决此问题，我们引入了$t$-截断Vendi分数，该方法通过截断核矩阵的特征谱，可证明在$n=\mathcal{O}(t)$个样本时保证收敛到其总体极限。我们进一步证明，现有的Nyström和FKEA近似方法收敛到截断Vendi分数的渐近极限。与Vendi分数相反，我们证明了RKE分数对所有核函数均具有普适的收敛保证。我们进行了多项数值实验，以说明通过Nyström和FKEA计算的Vendi分数围绕截断Vendi分数的集中程度，并分析了截断Vendi分数和RKE分数与图像及文本数据多样性的相关性。代码可在https://github.com/aziksh-ospanov/truncated-vendi获取。