  In a world abundant with diverse data arising from complex acquisition
techniques, there is a growing need for new data analysis methods. In this
paper we focus on high-dimensional data that are organized into several
hierarchical datasets. We assume that each dataset consists of complex samples,
and every sample has a distinct irregular structure modeled by a graph. The
main novelty in this work lies in the combination of two complementing powerful
data-analytic approaches: topological data analysis (TDA) and geometric
manifold learning. Geometry primarily contains local information, while
topology inherently provides global descriptors. Based on this combination, we
present a method for building an informative representation of hierarchical
datasets. At the finer (sample) level, we devise a new metric between samples
based on manifold learning that facilitates quantitative structural analysis.
At the coarser (dataset) level, we employ TDA to extract qualitative structural
information from the datasets. We showcase the applicability and advantages of
our method on simulated data and on a corpus of hyper-spectral images. We show
that an ensemble of hyper-spectral images exhibits a hierarchical structure
that fits well the considered setting. In addition, we show that our new method
gives rise to superior classification results compared to state-of-the-art
methods.
