Contrastive Functional Principal Components Analysis

As functional data assumes a central role in contemporary data analysis, the search for meaningful dimension reduction becomes critical due to its inherent infinite-dimensional structure. Traditional methods, such as Functional Principal Component Analysis (FPCA), adeptly explore the overarching structures within the functional data. However, these methods may not sufficiently identify low-dimensional representations that are specific or enriched in a foreground dataset (case or treatment group) relative to a background dataset (control group). This limitation becomes critical in scenarios where the foreground dataset, such as a specific treatment group in biomedical applications, contains unique patterns or trends that are not as pronounced in the background dataset. Addressing this gap, we propose Contrastive Functional Principal Component Analysis (CFPCA), a method designed to spotlight low-dimensional structures unique to or enriched in the foreground dataset relative to the background counterpart. We supplement our method with theoretical guarantees on CFPCA estimates supported by multiple simulations. Through a series of applications, CFPCA successfully identifies these foreground-specific structures, thereby revealing distinct patterns and trends that traditional FPCA overlooks.