Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction that is useful for various data science problems. However, many applications involve heterogeneous data that varies in quality due to noise characteristics associated with different sources of the data. Methods that deal with this mixed dataset are known as heteroscedastic methods. Current methods like HePPCAT make Gaussian assumptions of the basis coefficients that may not hold in practice. Other methods such as Weighted PCA (WPCA) assume the noise variances are known, which may be difficult to know in practice. This paper develops a PCA method that can estimate the sample-wise noise variances and use this information in the model to improve the estimate of the subspace basis associated with the low-rank structure of the data. This is done without distributional assumptions of the low-rank component and without assuming the noise variances are known. Simulations show the effectiveness of accounting for such heteroscedasticity in the data, the benefits of using such a method with all of the data versus retaining only good data, and comparisons are made against other PCA methods established in the literature like PCA, Robust PCA (RPCA), and HePPCAT. Code available at https://github.com/javiersc1/ALPCAH

翻译：主成分分析（PCA）是数据降维领域的关键工具，对各类数据科学问题具有重要价值。然而，许多应用涉及因数据来源不同而存在噪声特征差异的异质数据。处理这类混合数据集的方法被称为异方差方法。现有方法如HePPCAT对基系数的高斯假设在实际中可能不成立，而加权PCA等方法假设噪声方差已知，这在实践中往往难以实现。本文提出一种PCA方法，能够估计样本级噪声方差，并将其纳入模型以改进与数据低秩结构相关的子空间基估计。该方法无需对低秩分量进行分布假设，也无需假设噪声方差已知。仿真实验证明了考虑数据异方差性的有效性、在全部数据上使用该方法相较于仅保留优质数据的优势，并与PCA、鲁棒主成分分析及HePPCAT等现有PCA方法进行了对比。代码详见https://github.com/javiersc1/ALPCAH