Uniform Manifold Approximation & Projection (UMAP)

Updated 11 December 2025

UMAP is a nonlinear dimension reduction method that models local similarities using fuzzy simplicial sets from neighborhood graphs.
It optimizes a cross-entropy objective via stochastic gradient descent with negative sampling to balance local and global data structures.
UMAP is scalable and adaptable, supporting various metrics and extensions like parametric and approximate embeddings for diverse applications.

Uniform Manifold Approximation and Projection (UMAP) is a nonlinear dimension reduction and manifold learning technique designed for scalable visualization and representation of high-dimensional data. UMAP constructs a topological representation by modeling local similarities with fuzzy simplicial sets, then seeks a low-dimensional embedding that preserves these structures by optimizing a cross-entropy objective. It provides tunable trade-offs between local and global structure preservation, computational efficiency, and the ability to adapt to a wide array of dissimilarity metrics and data modalities.

1. Theoretical Foundations: Manifold Learning and Fuzzy Simplicial Sets

UMAP’s mathematical formulation is rooted in Riemannian geometry and algebraic topology. The method assumes the high-dimensional data $\{x_1,\ldots,x_n\} \subset \mathbb{R}^d$ is sampled from a manifold $\mathcal{M}$ with an unknown local metric $g$ . To approximate local geometry, UMAP rescale local distances by finding, for each $x_i$ , a local connectivity parameter $\rho_i$ (the distance to the closest neighbor) and a local smoothness/bandwidth $\sigma_i$ satisfying:

$\sum_{j: x_j \in \mathrm{NN}_k(i)} \exp\bigl(-\max(0, d_{ij} - \rho_i)/\sigma_i\bigr) = \log_2(k)$

where $k$ is the neighborhood size, and $d_{ij}$ is the chosen high-dimensional metric (often Euclidean, but alternatives are supported).

From these neighborhoods, UMAP constructs a weighted, directed $k$ -nearest neighbor graph, assigning directed fuzzy membership strength:

$\mu_{i|j} = \begin{cases} 1, & d_{ij} \leq \rho_i \ \exp\bigl(- (d_{ij} - \rho_i)/\sigma_i \bigr), & d_{ij} > \rho_i \end{cases}$

Symmetrization via probabilistic fuzzy union yields undirected edge weights:

$\mu_{ij} = \mu_{i|j} + \mu_{j|i} - \mu_{i|j} \cdot \mu_{j|i}$

Interpreting this weighted graph as the 1-skeleton of a fuzzy simplicial set, UMAP encodes local manifold structure suitable for downstream optimization (Chang, 16 Feb 2025, McInnes et al., 2018).

2. Objective Function and Optimization Dynamics

The low-dimensional embedding $\{y_1, \ldots, y_n\} \subset \mathbb{R}^{d'}$ is constructed so that its own fuzzy simplicial set best matches the high-dimensional one. Pairwise similarities in the target space are modeled by:

$\nu_{ij} = \frac{1}{1 + a\|y_i - y_j\|^{2b}}$

with $a$ , $b$ chosen by curve fitting to express a desired minimum inter-point distance (typical values: $a\approx1.929$ , $b\approx0.7915$ ).

The optimization minimizes cross-entropy between the two sets of affinities:

$C = \sum_{i \neq j} \Bigl[ \mu_{ij}\log\frac{\mu_{ij}}{\nu_{ij}} + (1-\mu_{ij})\log\frac{1-\mu_{ij}}{1-\nu_{ij}} \Bigr]$

or, equivalently (up to constants),

$C = \sum_{i \neq j} \left[ \mu_{ij}\log \nu_{ij} + (1-\mu_{ij})\log(1-\nu_{ij}) \right]$

UMAP employs stochastic gradient descent (SGD) with negative sampling: for each positive edge (neighbor pair with $\mu_{ij}>0$ ), several negative pairs (non-neighbors) are sampled to enforce repulsion. This leads to a non-convex optimization landscape, requiring careful tuning of step sizes and annealing schedules to ensure convergence and stability (Islam et al., 12 Mar 2025).

3. Algorithmic Pipeline, Hyperparameters, and Complexity

The algorithmic workflow for UMAP proceeds as follows:

Stage	Core Operations
High-dimensional graph construction	k-NN search, local entropy calibration, fuzzy set membership
Symmetrization	Fuzzy union (probabilistic t-norm) over directed memberships
Embedding initialization	Random draw, spectral method, or PCA
SGD optimization (w/ neg. sampling)	Attractive updates for neighbors, repulsive for random negatives

Critical hyperparameters:

n_neighbors (k): Controls locality/globality. Small $k$ emphasizes fine structure; large $k$ encourages global topology but may blur local features.
min_dist: Governs the minimum spacing in the embedding. Small values enable tight grouping; large values enforce even distribution.
metric: Choice of input space geometry (e.g., Euclidean, cosine, Manhattan), essential for meaningful neighborhood identification (Ashmead et al., 9 Dec 2025, Tucker et al., 2022).

With approximate NN-descent, neighbor graph construction is $O(n\log n)$ ; optimization is $O(nk\, n_\mathrm{epochs})$ with $k \ll n$ and moderate $n_\mathrm{epochs}$ ($200$–$500$), yielding practical scalability to millions of samples (Chang, 16 Feb 2025, Wei et al., 2020).

4. Variations, Extensions, and Computational Enhancements

UMAP is extensible along methodological, computational, and application-driven axes:

Variants with alternative input metrics: Swapping the Euclidean metric for problem-adapted measures (e.g., elastic shape distances) produces embeddings that respect intrinsic data invariances, critical for shape analysis and time-series (Tucker et al., 2022).
Parametric UMAP: Introduces a deep network $f_\theta(x)$ for the mapping, enabling batched updates and instant embedding of new/unseen points ( $z=f_\theta(x)$ ). This is crucial for integration into autoencoders, semi-supervised classifiers, and real-time pipelines (Sainburg et al., 2020, Ghojogh et al., 2021).
Approximate UMAP (aUMAP): After standard UMAP training, new points are embedded via weighted averaging of k-nearest neighbors in the original space. This provides order-of-magnitude speedups for streaming and online applications, trading small fidelity losses for reduced latency (Wassenaar et al., 5 Apr 2024).
Distributed and GPU-accelerated UMAP: Count Sketch–based approaches (“Sketch-and-Scale”) minimize memory and communication in geo-distributed contexts, while highly optimized GPU implementations (RAPIDS cuML) exploit fused kernels, memory pooling, and atomic reductions for up to 100 $\times$ speedups on large datasets (Nolet et al., 2020, Wei et al., 2020).
Preprocessing variants: Data-domain decompositions (e.g., correlated clustering and projection for scRNA-seq) serve as pre-UMAP dimensionality reduction, shaping the effective geometry and cluster separation in the embedding (Hozumi et al., 2023).

5. Analysis of Embedding Forces and Practical Tuning

The embedding process in UMAP is characterized by explicit attractive and repulsive forces derived from the gradient of the cross-entropy loss:

Attractive updates drive neighboring points together, while repulsive updates force random pairs apart. The force shapes,

$f_a^U(\zeta) = -\frac{2ab \zeta^{2(b-1)}}{1 + a\zeta^{2b}} \;\;\;\text{(attraction)}$

$f_r^U(\zeta) = \frac{2b}{\zeta^{2b} (1 + a\zeta^{2b})} \;\;\;\text{(repulsion)}$

govern cluster formation, inter-cluster spacing, and global layout (Islam et al., 12 Mar 2025). Notably, the attractive force can be expansive at very small distances, necessitating learning-rate annealing to prevent overshoot and oscillatory behavior. For enhanced consistency under random initialization, a linear tail can be added to the attraction to improve convergence to reproducible embeddings. Manipulating these force shapes—by adjusting $a$ , $b$ , or adding global attraction/repulsion—offers direct control over the compactness, separation, and stability of clusters in the output space.

6. Empirical Performance and Domain Applications

UMAP has established itself as a leading tool for visualization, exploratory analysis, and representation learning across multiple domains:

Visualization of complex manifolds: UMAP recovers both distinct clusters (e.g., digits in MNIST, classes in single-cell RNA-seq) and global topological features with robustness to subsampling (McInnes et al., 2018, Ghojogh et al., 2021).
Quantitative improvements: On classification and clustering tasks, UMAP often matches or exceeds t-SNE in local neighborhood preservation, while providing superior global structure retention and significantly faster runtimes. In scRNA-seq, CCP-UMAP improves ARI by ~19% over standard UMAP, while in astrophysical color–redshift mapping, UMAP-based regression yields lower outlier rates and improved fidelity relative to SOM grids (Hozumi et al., 2023, Ashmead et al., 9 Dec 2025).
Scalability: Leveraging ANN-Descent for neighbor search and negative sampling optimization, UMAP scales to millions of data points (e.g., SDSS stars, hyperspectral imaging) with cost-effective memory/communication usage, especially with sketching or GPU implementations (Nolet et al., 2020, Wei et al., 2020).
Real-time/online adaptation: aUMAP and parametric UMAP enable high-rate online embedding suitable for streaming data and model introspection in dynamic settings (e.g., BCI signals) (Wassenaar et al., 5 Apr 2024).

7. Strengths, Limitations, and Method Selection

UMAP’s core strengths include:

Scalability via $O(n \log n)$ construction/optimization, with tuning options for target fidelity, cluster compactness, and global topology (Chang, 16 Feb 2025).
Flexibility in metric choice, accommodating problem-driven geometry (e.g., shape-invariant, elastic, or custom distances) (Tucker et al., 2022).
Expressive embeddings that can be incorporated as regularizers in downstream deep-learning pipelines (autoencoders, semi-supervised classifiers) (Sainburg et al., 2020).

Limitations:

Absence of an analytic inverse transform; mapping embedded points back to the original space is non-trivial (Chang, 16 Feb 2025).
Sensitivity to hyperparameter settings; improper calibration of $k$ or min_dist can fragment or oversmooth the manifold.
Non-convexity of the loss; stochasticity in initialization and optimization may lead to variability in embeddings.
For settings where interpretability via explicit loadings or projection axes is required, or pure linearity is sufficient, PCA/KPCA may be more appropriate.

A practical workflow often leverages UMAP for rapid manifold discovery and initial visualization, followed by clustering or quantitative analysis in the low-dimensional space.

In summary, UMAP provides a mathematically rigorous, scalable, and versatile framework for nonlinear dimensionality reduction. By faithfully approximating manifold geometry via fuzzy simplicial sets and optimizing a cross-entropy objective, it achieves high-quality embeddings with rich structure preservation, broad applicability across domains, and adaptability to contemporary computational requirements (Chang, 16 Feb 2025, Ashmead et al., 9 Dec 2025, Wassenaar et al., 5 Apr 2024, Sainburg et al., 2020, Hozumi et al., 2023).