研究多概率测度的PCA收敛性，揭示稀疏与密集采样间的转换规律。
PCA of probability measures: Sparse and Dense sampling regimes
- 在双渐近框架下分析n个测度各由m样本观测时的PCA收敛性
- 得到误差率n⁻¹/² + m⁻ᵃ，α由嵌入方式决定，揭示采样密度影响
- 证明密集采样下率可达极小最大最优，适合高维概率数据分析
对概率测度进行主成分分析（PCA）的常见方法是将其嵌入希尔伯特空间，从而应用标准函数型PCA技术。尽管从m个样本估计单个测度的嵌入收敛速率已有充分研究，但关于多个测度的情形尚无系统分析。本文研究在双渐近框架下的PCA：共观测n个概率测度，每个测度通过m个样本获得。我们推导出经验协方差算子和PCA过拟合风险的收敛率形式为n⁻¹/² + m⁻ᵃ，其中α>0依赖于所选嵌入方式。该结果刻画了测度数量n与每测度样本数m之间的关系，揭示了从稀疏（小m）到密集（大m）采样模式的收敛行为转变。此外，我们证明密集情形下的误差率对经验协方差误差达到极小最大最优。数值实验验证了这些理论结果，并表明合理子采样可在降低计算成本的同时保持PCA精度。
原文摘要 · Abstract (English)
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.
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