  Principal component analysis (PCA) is a classical and ubiquitous method for
reducing data dimensionality, but it is suboptimal for heterogeneous data that
are increasingly common in modern applications. PCA treats all samples
uniformly so degrades when the noise is heteroscedastic across samples, as
occurs, e.g., when samples come from sources of heterogeneous quality. This
paper develops a probabilistic PCA variant that estimates and accounts for this
heterogeneity by incorporating it in the statistical model. Unlike in the
homoscedastic setting, the resulting nonconvex optimization problem is not
seemingly solved by singular value decomposition. This paper develops a
heteroscedastic probabilistic PCA technique (HePPCAT) that uses efficient
alternating maximization algorithms to jointly estimate both the underlying
factors and the unknown noise variances. Simulation experiments illustrate the
comparative speed of the algorithms, the benefit of accounting for
heteroscedasticity, and the seemingly favorable optimization landscape of this
problem.
