Seminar talk by Prof. C. G. Small

Dimensionality reduction and manifold learning techniques attempt to recover a lower dimensional submanifold from the data as encoded in high-dimensions. Standard methods, such as Isomap and LLE, map the high dimensional data points into low dimension so as to globally minimize a so-called energy function, which measures the mismatch between the precise geometry in high dimensions and the approximate geometry in low dimensions. In this talk, we introduce a new algorithm for nonlinear dimensionality reduction called Spanifold which constructs a tree on the manifold, and flattens the manifold in such a way as to approximately preserve pairwise distance relationships within the tree.  The vertices of this tree are the data points, and the edges of the tree form a subset of the edges of nearest neighbour graph on the data.  In addition, the pairwise distances between data points close to the root of the tree undergo minimal distortionas the data are flattened.

This allows the user to design the flattening algorithm so as to approximately preserve neighbour relationships inany chosen local region of the data. How to assess or compare the performances of different methods is still a open problem. It is known that some commonly used methods are sensitive to the presence of outliers, or the violation of model assumptions. The robustness of dimensionality reduction methods is another problem that not yet well explored. In this talk we also establish some criteria to address these issues. A goodness measure called local Spearman correlation is developed to assess the performance of dimensionality reduction methods. Based on the goodness measure, a type of influence function and breakdown point are defined to study the robustness of dimensionality reduction methods. (This is joint work with S. Chenouri, J. Liang and P. Kobelevskiy.)