# day 3

$$\def\Real{\mathbb{R}}$$

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.
• PCA;
• nonlinear PCA.
• Model-free tools
• Multidimensional scaling. Low rank matrix approximations are an important tool for MDS.
• Isomap.
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.