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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