Spectral Methods for Single-cell, Spatial, and Multi-omics Data

Abstract

In the past decade new biological technologies have made it possible to measure genomic quantities at the resolution of individual cells. However, scientific discovery using data from these technologies is challenging due to the vast size and complex noise. This dissertation presents three statistical methods motivated by biological problems from new sequencing technologies. In the first chapter, we develop a generalized bilinear model to perform dimensionality reduction for single-cell RNA-seq data. By directly modeling the data using a count distribution, our method is able to overcome limitations of existing approaches and provide biologists an unbiased view into the cell types present in their data. In the second chapter, we consider spatially resolved single-cell data and design an approach to estimate a curve that accurately models the structure of one-dimensional tissue. By leveraging this estimated curve, we are able to identify biologically relevant spatially variable genes that were missed by existing approaches. In the final chapter, we study two widely-used approaches for estimating shared structure from multiple genomic data matrices (``multi-omics"). By leveraging new results in random matrix theory, we precisely characterize the tradeoffs between these approaches and design extensions that leads to more powerful data integration. Across all chapters, spectral decompositions of problem-specific matrices provide a unifying framework for extracting structure from high-dimensional data.