\(\def\Real{\mathbb{R}}\)

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.

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