A study on validating non-linear dimensionality reduction using persistent homology

During the process of non-linear dimensionality reduction, manifolds represented by point clouds are at risk of changing their topology. We review techniques for quality assessment of manifold learning and propose to use persistent homology to evaluate the topological impact of manifold learning by comparing the Betti numbers of test manifolds before and after dimensionality reduction. We propose a benchmark suite of test manifolds based on the Swiss roll dataset with added geometrical and topological complexity. The experiments demonstrate the effectivity of the approach by analysing examples of test manifolds where the embedding failed. Betti numbers based on persistent homology are also used to select suitable sampling rates for the manifold point clouds and to determine optimal values for the nearest neighbour parameter k of selected manifold learning methods. The results indicate that the more complex the manifold is the more sample points and larger values for k are required.