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Dimensionality reduction has become one of the fundamental problems in modern mathematical data science. Standard methods for embedding data or dimension reduction include t-SNE [15], PCA [21], UMAP [16] and many others. There is an interesting and important gap between methods with solid theoretical footing (e.g., PCA and SVD, multi-dimensional scaling, and spectral methods) and methods which are less well understood albeit widely used in practice (e.g., t-SNE or UMAP). We focus on t-SNE (which is closely related to UMAP [4, 12]) because it is widely used for data visualization, dimensionality reduction, and clustering, yet it lacks extensive theoretical analysis. Because this method is so effective for visualizing well-clustered, high-dimensional data, it has become an indispensable tool for biological data analysis tasks, including, but not limited to single-cell transcriptomics [10], single-cell flow, mass cytometry [1, 19], and whole-genome sequencing [13, 5].