Manifold Learning and Dimensionality Reduction with Diffusion Maps

  1. Introduction
    1. Principal Component Analysis
    2. Graph-Based Algorithms
    3. Locally Linear Embedding
    4. Isomap
    5. Laplacian Eigenmaps
      1. Graph Laplacians
      2. The Algorithm
      3. Algorithm Justification
      4. Eigenmaps - Conclusion
  2. Diffusion Maps
    1. Intuition
    2. Diffusion Distance
    3. Embedding
    4. Conclusion
  3. Anisotropic Diffusion
    1. Family of Anisotropic Diffusions
    2. Laplace-Beltrami Operator
    3. Influence of Density and Geometry
  4. Conclusion
  5. Appendix:
    1. PCA code and mapping to Eigenvector
    2. Rayleigh Ritz Proof
    3. Implementation of Laplacian Eigenmaps
    4. Implementation of Anisotropic Diffusion
    5. Implementation of Eigenfunctions for Symmetric and Random Walk Laplacian