Probabilistic PCA for Heteroscedastic Data

David Hong, Laura Balzano, Jeffrey A. Fessler

Abstract: Principal Component Analysis (PCA) is a standard dimensionality reduction technique, but it treats all samples uniformly' making it suboptimal for heterogeneous data that are increasingly common in modern settings. This paper proposes a PCA variant for samples with heterogeneous noise levels, i.e., heteroscedastic noise, that naturally arise when some of the data come from higher quality sources than others. The technique handles heteroscedasticity by incorporating it in the statistical model of a probabilistic PCA. The resulting optimization problem is an interesting nonconvex problem related to but not seemingly solved by singular value decomposition, and this paper derives an expectation maximization (EM) algorithm. Numerical experiments illustrate the benefits of using the proposed method to combine samples with heteroscedastic noise in a single analysis, as well as benefits of careful initialization for the EM algorithm.