Using persistent homology to understand dimensionality reduction in resting-state fMRI

Evaluating the success of a manifold learning method remains a challenging problem, especially for methods adapted to a specific application domain. The present work investigates shared geometric structure across different dimensionality reduction (DR) algorithms within the scope of neuroimaging applications. We examine reduced-dimension embeddings produced by a representative assay of dimension reductions for brain data ("brain representations") through the lens of persistent homology, making statistical claims about topological differences using a recent topological boostrap method. We cluster these methods based on their induced topologies, finding feature type and number -- rather than reduction algorithm -- as the main drivers of observed topological differences.