Abstract: This paper systematically introduces the theoretical foundations,algorithmic implementations,and recent developments of UMAP algorithm.UMAP is a nonlinear dimensionality reduction method grounded in algebraic topology and category theory.It aims to preserve the local structure of high-dimensional data while faithfully capturing its global distribution.The algorithm can be divided into two stages.Firstly,it constructs a weighted k-nearest neighbor graph that encodes local geometric relationships,with neighborhood sizes adaptively determined by data density.Secondly,it learns a low-dimensional embedding by minimizing the cross-entropy between the high-dimensional and low-dimensional neighborhood graphs,thereby preserving topological structure and enabling effective visualization.This survey further reviews several notable extensions of UMAP.1) Supervised and Semi-supervised UMAP enhances class separation by incorporating label information;2) Parameterized UMAP integrates neural networks to achieve generalizable nonlinear mappings;3) DensMAP preserves data distribution characteristics through density correlation optimization;4) AlignedUMAP enables aligned embeddings across datasets or temporal sequences;5) Progressive UMAP addresses dynamic embedding challenges for streaming data and out-of-sample extensions;6)GNUMAP,which combines with graph neural networks;7)MultiMAP,which is applied to multimodal data fusion.

Key words: UMAP, Supervised learning, Semi-supervised learning, Parameterized UMAP, DensMAP, AlignedUMAP, Progressive UMAP, GNUMAP, MultiMAP