day 3

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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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