Characterizing non-linear dimensionality reduction methods using Laplacian-like operators

(note: the talk is 30 mins, but the server has problems with 12:00 noon)

We examine a number of non-linear dimensionality reduction techniques including Laplacian Eigenmaps, LLE, MVU, HLLE, LTSA, and t-SNE. In each case we show that the non-linear embedding can be characterized by a Laplacian or Laplacian-like operator. By comparing the resulting operators, one can uncover the similarities and differences between the methods. For example, HLLE and LTSA can be shown to be asymptotically identical, and whilst maximum variance unfolding (MVU) can be shown to generate a Laplacian, the behavior of the Laplacian is completely different from that generated by Laplacian Eigenmaps. We discuss the implications of this characterization for generating new non-linear dimensionality reduction methods and smoothness penalties.