Single-cell transcriptomics has revolutionized cancer research, allowing us to characterize intratumoral heterogeneity and continuous developmental trajectories. Dimensionality reduction techniques, specifically t-Distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP), have become the standard for visualizing these high-dimensional manifolds. However, while these algorithms are successfully used as investigative tools, their mathematical foundations regarding global structure preservation, stability, and interpretability remain fragile. In this talk, I will highlight these specific open problems.
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