  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.
