  Over the years, Principal Component Analysis (PCA) has served as the baseline
approach for dimensionality reduction in gene expression data analysis. It
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 classification 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 improved PCA
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
incorporate higher-order simplicial complexes to capture higher-order
interactions in the data. We evaluate and validate the performance of PLPCA
using benchmark microarray datasets that involve normal tissue samples and four
different cancer tissues. Our extensive studies demonstrate that PLPCA
outperforms all other state-of-the-art models for classification tasks after
dimensionality reduction.
