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Abstract:
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Principal Components Analysis (PCA) is a popular technique for revealing structure in data. However, in practical applications the PCs are often hard to interpret, and in high dimensions can be unreliable. One way to improve both interpretability and reliability of PCs is to assume that they are sparse. Most sparse PCA methods rely on sparsity-inducing penalties with tuning parameters that control the amount of sparsity. Because sparsity is expected to vary among PCs, it is desirable to have a separate tuning parameter for each PC. However, properly tuning multiple parameters by standard methods of cross-validation (CV) is computationally challenging. We call this the "Multiple Tuning Problem" (MTP). Here we introduce an empirical Bayes approach to sparse PCA that provides a principled and efficient way to solve the MTP. Our method posits a (sparse) prior distribution for each PC, whose hyperparameters are learned by maximum likelihood, through a computationally-efficient MM (minorize-maximize) algorithm, rather than CV. We demonstrate the effectiveness of this approach by simulation examples.
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