Schubert variety of best fit with applications and across domains sparse feature extraction

Abstract

This dissertation presents two novel approaches in applied mathematics for data analysis and feature selection, addressing challenges in both geometric data representation and multi-domain biological data interpretation. The first part introduces the Schubert Variety of Best Fit (SVBF) as a new geometric framework for analyzing sets of datasets. Leveraging the structure of Grassmann manifolds and Schubert varieties, we develop the SVBF-Node, a computational unit for solving related optimization problems. We demonstrate the efficacy of this approach through three classification algorithms and a new clustering method, SVBF-LBG. These techniques are evaluated on various datasets, including synthetic data, image sets, video sequences, and hyperspectral remote sensing data, showing improved performance over existing similar methods, particularly for complex, high-dimensional data. The second part proposes a multi-domain, multi-task (MDMT) architecture for feature selection in biological data. This method integrates multi-domain learning with masked feature selection, specifically applied to gene expression data from multiple tissues. We demonstrate its ability to identify novel biomarkers in host immune responses to infection, which are not detectable through single-domain analyses. The approach is validated using bulk RNA sequences from different tissues, revealing its potential to uncover cross-domain biological insights. Both contributions offer interpretable, mathematically grounded approaches to data analysis, providing new tools for researchers in applied mathematics, machine learning, and bioinformatics.