Introduction:

• Overview of manifolds and the basic topology of data
• Statistical learning and instrinsic dimensionality
• The manifold hypothesis

Chapter 1: Multidimensional Scaling

• Classical, metric, and non-metric MDS algorithms
• Example applications to quantitative psychology and social science

Chapter 2: ISOMAP

• Geodesic distances and the isometric mapping algorithm
• Implementation details and applications with facial images and coil-100 object images

Chapter 3: Local Linear Embedding

• Locally linear reconstructions and optimization problems
• Example applications with image data

Chapter 4: Laplacian Eigenmaps/Spectral Embedding

• From the general to the discrete Laplacian operators
• Visualizing spectral embedding with the networkx library
• Spectral embedding with NLTK and the Brown text corpus