UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. It is constructed from a theoretical framework based on Riemannian geometry and algebraic topology, resulting in a practical and scalable algorithm suitable for real-world data. UMAP is competitive with t-SNE in visualization quality and preserves more of the global structure with superior runtime performance. It has no computational restrictions on embedding dimension, making it a viable general-purpose dimension reduction technique for machine learning. The paper introduces UMAP, providing a sound mathematical theory and a practical scalable algorithm. UMAP addresses the issue of uniform data distributions on manifolds through a combination of Riemannian geometry and category-theoretic approaches to geometric realization of fuzzy simplicial sets. The algorithm optimizes the layout of data in a low-dimensional space to minimize the cross-entropy between the topological representations of the high-dimensional and low-dimensional data. The theoretical foundations of UMAP are rooted in manifold theory and topological data analysis. The algorithm constructs local manifold approximations and patches together their local fuzzy simplicial set representations to form a topological representation of the high-dimensional data. It then optimizes the layout of the data in the low-dimensional space to minimize the cross-entropy between the topological representations. The computational description of UMAP involves constructing a weighted k-neighbour graph and performing a force-directed graph layout. The algorithm uses stochastic gradient descent to optimize the embedding, with a focus on minimizing the fuzzy set cross entropy between the weighted graphs. The implementation details include efficient approximate k-nearest-neighbor computation and optimization via stochastic gradient descent. The overall complexity is bounded by the approximate nearest neighbor search complexity, which is empirically approximately \(O(N^{1.14})\). The paper discusses the effects of hyper-parameters, such as the number of neighbors \(n\) and the desired separation between close points in the embedding space, on the performance of UMAP.UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. It is constructed from a theoretical framework based on Riemannian geometry and algebraic topology, resulting in a practical and scalable algorithm suitable for real-world data. UMAP is competitive with t-SNE in visualization quality and preserves more of the global structure with superior runtime performance. It has no computational restrictions on embedding dimension, making it a viable general-purpose dimension reduction technique for machine learning. The paper introduces UMAP, providing a sound mathematical theory and a practical scalable algorithm. UMAP addresses the issue of uniform data distributions on manifolds through a combination of Riemannian geometry and category-theoretic approaches to geometric realization of fuzzy simplicial sets. The algorithm optimizes the layout of data in a low-dimensional space to minimize the cross-entropy between the topological representations of the high-dimensional and low-dimensional data. The theoretical foundations of UMAP are rooted in manifold theory and topological data analysis. The algorithm constructs local manifold approximations and patches together their local fuzzy simplicial set representations to form a topological representation of the high-dimensional data. It then optimizes the layout of the data in the low-dimensional space to minimize the cross-entropy between the topological representations. The computational description of UMAP involves constructing a weighted k-neighbour graph and performing a force-directed graph layout. The algorithm uses stochastic gradient descent to optimize the embedding, with a focus on minimizing the fuzzy set cross entropy between the weighted graphs. The implementation details include efficient approximate k-nearest-neighbor computation and optimization via stochastic gradient descent. The overall complexity is bounded by the approximate nearest neighbor search complexity, which is empirically approximately \(O(N^{1.14})\). The paper discusses the effects of hyper-parameters, such as the number of neighbors \(n\) and the desired separation between close points in the embedding space, on the performance of UMAP.