Open Mathematical Problems in Manifold Learning for Single-Cell Data

Explore open mathematical challenges in manifold learning techniques used for single-cell data analysis in this 51-minute conference talk from IPAM's Mathematics of Cancer workshop. Delve into how single-cell transcriptomics has transformed cancer research by enabling characterization of intratumoral heterogeneity and continuous developmental trajectories. Examine the widespread use of dimensionality reduction techniques, particularly t-Distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP), as standard tools for visualizing high-dimensional biological manifolds. Investigate the mathematical foundations underlying these algorithms, focusing on critical gaps in understanding global structure preservation, stability, and interpretability that remain despite their successful application as investigative tools. Gain insights into specific unsolved problems that represent important frontiers in the mathematical analysis of single-cell data visualization methods.