Joint Geometric and Topological Analysis of Hierarchical Datasets
This work addresses the need for new data analysis methods for complex hierarchical datasets, such as hyper-spectral images, but it appears incremental as it combines existing approaches (TDA and manifold learning) rather than introducing a fundamentally new paradigm.
The paper tackles the problem of analyzing high-dimensional hierarchical datasets with irregular graph-structured samples by combining topological data analysis and geometric manifold learning, resulting in a new method that yields superior classification results compared to state-of-the-art methods on simulated data and a corpus of hyper-spectral images.
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.