PP: UMAP: uniform manifold approximation and projection for dimension reduction

Posted 2021-03-13 渡轮终会回航

From Tutte institute for mathematics and computing

Problem: dimension reduction

Theoretical foundations:

Methodology:

Two phases.

In the first phase, a particular weighted k-neighbour graph is constructed. In the second phase, a low dimensional layout of this graph is computed

 

 

 

Supplementary knowledge:

1. simplicial sets. 单纯集

In mathematics, a simplicial set is an object made up of "simplices单纯形" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. 

simplex: 单纯形, 

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron四边形 to arbitrary dimensions.

For example,

• a 0-simplex is a point,
• a 1-simplex is a line segment,
• a 2-simplex is a triangle,
• a 3-simplex is a tetrahedron,
• a 4-simplex is a 5-cell.

2. Hadamard product/ pointwise product

3.