CONTRASTIVE DIMENSION REDUCTION: METHODS FOR FUNCTIONAL DATA, FEATURE AND SAMPLE SELECTION, AND MULTI-TREATMENT ANALYSIS

Advances in technology have enabled the collection of high-dimensional data across many scientific domains, creating a growing need for methods that can extract meaningful structure from complex datasets. Dimension reduction (DR) techniques address this challenge by aiding visualization, interpretation, and downstream analysis. However, when data contain multiple groups (e.g., case-control), traditional DR methods often fail to capture group-specific signals. This area is known as contrastive dimension reduction. This dissertation develops three contrastive DR methodologies to identify structure enriched in one group relative to others: contrastive functional dimension reduction, contrastive feature and sample selection, and contrastive liquid association. Chapter 2 introduces contrastive functional principal component analysis (CFPCA), a method for functional data that extracts low-dimensional representations enriched in a target group. While contrastive PCA (CPCA) is effective for multivariate data, it does not directly extend to functional settings. CFPCA addresses this limitation by discretizing functional observations and contrasting covariance structures between groups. Inspired by contrastive latent variable modeling (CLVM), simulations and real-data applications demonstrate that CFPCA better isolates contrastive structure than traditional functional PCA (FPCA). Chapter 3 presents contrastive CUR (CCUR), extending the CUR matrix decomposition to the contrastive setting. Using leverage scores, CCUR selects features and samples that are most informative for distinguishing groups. Applications to protein expression and single-cell RNA sequencing (scRNA-seq) data show that CCUR identifies biologically meaningful patterns while maintaining interpretability. Chapter 4 considers settings with more than two groups, where existing contrastive methods are limited. By connecting CPCA with liquid association (LA), we develop a new framework that captures weighted contrasts across multiple treatments. This approach provides a principled summary of how treatments differ from control and supports visualization and feature selection. Chapter 5 concludes with a discussion of implications and future research directions.