A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-α}$ for the empirical covariance operator and the PCA excess risk, where $α>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.

翻译：对概率测度进行主成分分析的一种常见方法是将它们嵌入到希尔伯特空间中，从而应用标准函数型主成分分析技术。虽然从 $m$ 个样本估计单个测度嵌入的收敛速率已有充分研究，但现有文献尚未涉及涉及多个测度的场景。本文研究了一种双渐近机制下的主成分分析，其中观测到 $n$ 个概率测度，每个测度通过 $m$ 个样本获得。我们推导了经验协方差算子与主成分分析超额风险的收敛速率，其形式为 $n^{-1/2} + m^{-α}$，其中 $α>0$ 取决于所选嵌入方式。这一结果刻画了测度数量 $n$ 与每个测度样本数 $m$ 之间的关系，揭示了收敛行为中从稀疏（$m$ 较小）到密集（$m$ 较大）的转变。此外，我们证明了密集机制下的速率对于经验协方差误差是极小极大最优的。数值实验验证了这些理论速率，并表明适当的子采样能在降低计算成本的同时保持主成分分析的准确性。