Thom’s strong transversality theorem. Jets.
Examples: codimension of set of points of given corank.Typical singularities of visible contours.
Taken’s embedding theorem.
Short survey of geometric dimensionality reduction tools in data analysis.
- Model-based dimensionality reduction tools.
Setup: a point cloud in high-dimensional space. Problem: recover the underlying low-dimensional model.
- Random projections – parsimonious tool, does not recover the model though.
- Special projections. Model: linear or polynomial variety, assumed, not recovered.
- nonlinear PCA.
- Model-free tools
- Multidimensional scaling. Low rank matrix approximations are an important tool for MDS.
Create a network from the proximity data; use the functions measuring distance to a point to embed the point cloud into high-dimensional space, project to low-dimensional space.
- Eigenmap, diffusion map.
Create the network using the proximity data, – in case of diffuction maps, add weights, – and use the eigenvectors of the Laplacian to embed the space into some highdimensional Euclidean space. Then find low-dimensional projection.