Entrywise Recovery Guarantees for Sparse PCA via Sparsistent Algorithms

An $\ell_p$ theory of PCA and spectral clustering

Principal Component Analysis (PCA) is a powerful tool in statistics and machine learning. While existing study of PCA focuses on the recovery of principal components and their associated eigenvalues, there are few precise characterizations of individual principal component scores that yield low-dimensional embedding of samples. That hinders the analysis of various spectral methods. In this paper, we first develop an $\ell_p$ perturbation theory for a hollowed version of PCA in Hilbert spaces which provably improves upon the vanilla PCA in the presence of heteroscedastic noises. Through a novel…

超越正弦平方误差：流式主成分分析的线性时间逐项不确定性量化

Beyond Sin-Squared Error: Linear-Time Entrywise Uncertainty Quantification for Streaming PCA

本文针对流式PCA中估计特征向量逐项不确定性量化缺失的问题，基于Oja算法推导了逐项Bernstein型集中界和中心极限定理，并提出一种利用中位数均值的子采样方差估计算法，在保证统计一致性的同时显著降低计算成本。

具有缺失数据的异方差 PCA 推断

Inference for Heteroskedastic PCA with Missing Data

针对高维 PCA 中缺失数据与异方差噪声导致的不确定性量化难题，论文基于 HeteroPCA 建立非渐近分布保证，提出数据驱动的主子空间置信区域和尖峰协方差矩阵逐元素置信区间方法。

多元函数数据异常检测的低秩与结构化稀疏张量分解

Low-Rank and Structured Sparse Tensor Decomposition for Anomaly Detection in Multivariate Functional Data

针对多元函数数据异常检测，提出ES-CP和FG-Lasso两种无监督稀疏张量分解方法，通过低秩CP分解和结构化稀疏惩罚，在模拟与锻造案例中显著提升检测与定位性能。

随机化奇异值分解的扰动分析及其在统计学中的应用

Perturbation Analysis of Randomized SVD and its Applications to Statistics

本文对随机化奇异值分解（RSVD）进行扰动分析，推导左奇异向量及其投影的误差上界，并应用于子空间估计、社区检测、矩阵补全和缺失数据PCA，获得近乎最优收敛速率与渐近正态性。