PLPCA: Persistent Laplacian-Enhanced PCA for Microarray Data Analysis.

Over the years, Principal Component Analysis (PCA) has served as the baseline approach for dimensionality reduction in gene expression data analysis. Its primary objective is to identify a subset of disease-causing genes from a vast pool of thousands of genes. However, PCA possesses inherent limitations that hinder its interpretability, introduce class ambiguity, and fail to capture complex geometric structures in the data. Although these limitations have been partially addressed in the literature by incorporating various regularizers, such as graph Laplacian regularization, existing PCA based methods still face challenges related to multiscale analysis and capturing higher-order interactions in the data. To address these challenges, we propose a novel approach called Persistent Laplacian-enhanced Principal Component Analysis (PLPCA). PLPCA amalgamates the advantages of earlier regularized PCA methods with persistent spectral graph theory, specifically persistent Laplacians derived from algebraic topology. In contrast to graph Laplacians, persistent Laplacians enable multiscale analysis through filtration and can incorporate higher-order simplicial complexes to capture higher-order interactions in the data. We evaluate and validate the performance of PLPCA using ten benchmark microarray data sets that exhibit a wide range of dimensions and data imbalance ratios. Our extensive studies over these data sets demonstrate that PLPCA provides up to 12% improvement to the current state-of-the-art PCA models on five evaluation metrics for classification tasks after dimensionality reduction.