Uniform Manifold Approximation and Projection (UMAP)

UMAP begins by constructing a weighted k-neighbor graph representing the data's topology in high-dimensional space. It models the manifold as a fuzzy simplicial set—a mathematical object that captures both local connectivity and uncertainty. For each cell, UMAP computes a radius that ensures uniform manifold coverage, adapting to varying data densities. This step preserves the local neighborhood structure that defines cell-type relationships in single-cell data.

UMAP optimizes a low-dimensional embedding by minimizing the cross-entropy between the high-dimensional fuzzy simplicial set and its low-dimensional counterpart. This loss function has two distinct components:

• Attractive forces: Pull together cells that are neighbors in high-dimensional space, preserving local structure.
• Repulsive forces: Push apart cells that are not neighbors, preserving global separation between distinct cell types. This balance prevents the crowding problem that plagues t-SNE.

Unlike t-SNE, which focuses almost exclusively on local neighborhoods, UMAP preserves meaningful global distances between clusters. In single-cell data, this means the relative positioning of major cell lineages—such as the distance between T-cells and B-cells—carries biological meaning. UMAP achieves this through its repulsive loss term and by initializing the low-dimensional layout using spectral embedding, which captures the coarse global topology before local refinement begins.

UMAP supports supervised dimensionality reduction by incorporating target labels into the simplicial set construction. When cell-type annotations or experimental conditions are available, UMAP can use them to:

• Increase the effective distance between cells of different classes.
• Tighten clusters within the same class. This produces embeddings where known biological groupings are more clearly separated, aiding in comparative analysis across conditions or time points.

UMAP is designed for linear computational scaling with dataset size, making it suitable for modern single-cell atlases containing millions of cells. Key performance characteristics:

• Stochastic gradient descent with negative sampling accelerates optimization.
• Sparse nearest-neighbor graphs reduce memory overhead.
• GPU-accelerated implementations (e.g., Rapids cuML) achieve sub-second embedding times for large datasets. This scalability enables interactive exploration of entire organism-level cell atlases.

UMAP is grounded in Riemannian geometry and algebraic topology. It assumes that high-dimensional data lies on a low-dimensional manifold and constructs a custom Riemannian metric that is locally constant on each neighborhood. This theoretical foundation provides:

• Formal guarantees about the preservation of topological structure.
• A principled framework for extending UMAP to new data types and distance metrics.
• The ability to embed into arbitrary target dimensions (not just 2D or 3D) for downstream machine learning tasks.