# Uniform Manifold Approximation and Projection (UMAP) and its Variants: Tutorial and Survey

arXiv: https://arxiv.org/abs/2109.02508 · HTML: https://mlsanity.com/paper/2109.02508/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 34 references that are in the map.
- First posted on arXiv 25 Aug 2021.

## Builds on (most cited first, 9 of 9)

- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Visualizing and Measuring the Geometry of BERT](https://mlsanity.com/paper/1906.02715/) (2019)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [SenseBERT: Driving Some Sense into BERT](https://mlsanity.com/paper/1908.05646/) (2019)
- [On UMAP's true loss function](https://mlsanity.com/paper/2103.14608/) (2021)
- [Fisher Discriminant Triplet and Contrastive Losses for Training Siamese Networks](https://mlsanity.com/paper/2004.04674/) (2020)
- [Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Survey](https://mlsanity.com/paper/2106.02154/) (2021)
- [Stochastic Neighbor Embedding with Gaussian and Student-t Distributions: Tutorial and Survey](https://mlsanity.com/paper/2009.10301/) (2020)

## Built on by (most cited first, 10 of 10)

- [Tree Variational Autoencoders](https://mlsanity.com/paper/2306.08984/) (2023)
- [Abstractive Summarization of Large Document Collections Using GPT](https://mlsanity.com/paper/2310.05690/) (2023)
- [Balancing between the Local and Global Structures (LGS) in Graph Embedding](https://mlsanity.com/paper/2308.16403/) (2023)
- [Expert-Agnostic Ultrasound Image Quality Assessment using Deep Variational Clustering](https://mlsanity.com/paper/2307.02462/) (2023)
- [TuckA: Hierarchical Compact Tensor Experts for Efficient Fine-Tuning](https://mlsanity.com/paper/2511.06859/) (2025)
- [Attraction-Repulsion Swarming: A Generalized Framework of t-SNE via Force Normalization and Tunable Interactions](https://mlsanity.com/paper/2411.10617/) (2024)
- [Detecting Suspicious Commenter Mob Behaviors on YouTube Using Graph2Vec](https://mlsanity.com/paper/2311.05791/) (2023)
- [Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reduction](https://mlsanity.com/paper/2512.03899/) (2025)
- [A Diffusion-Contrastive Graph Neural Network with Virtual Nodes for Wind Nowcasting in Unobserved Regions](https://mlsanity.com/paper/2604.10328/) (2026)
- [Audience Engagement with Arabic Women's Social Empowerment and Wellbeing: A Decadal Corpus](https://mlsanity.com/paper/2605.22204/) (2026)

## Abstract (arXiv)

Uniform Manifold Approximation and Projection (UMAP) is one of the state-of-the-art methods for dimensionality reduction and data visualization. This is a tutorial and survey paper on UMAP and its variants. We start with UMAP algorithm where we explain probabilities of neighborhood in the input and embedding spaces, optimization of cost function, training algorithm, derivation of gradients, and supervised and semi-supervised embedding by UMAP. Then, we introduce the theory behind UMAP by algebraic topology and category theory. Then, we introduce UMAP as a neighbor embedding method and compare it with t-SNE and LargeVis algorithms. We discuss negative sampling and repulsive forces in UMAP's cost function. DensMAP is then explained for density-preserving embedding. We then introduce parametric UMAP for embedding by deep learning and progressive UMAP for streaming and out-of-sample data embedding.