Gromov-Hausdorff limit of Wasserstein spaces on point clouds

The resulting graph structure can then be used to design procedures for data clustering (unsupervised learning), data classification (supervised learning) and dimensionality reduction. For all of these graph-based procedures, it is important (as with any procedure in statistics) to study their consistency and to quantify how accurately they reveal features from the ground-truth distribution. Many of the graph-based procedures for which there are consistency results available are really optimization problems whose objective functionals incorporate the graph structure in one way or the other. This is the case for procedures like spectral clustering and total variation clustering, where the notions of graph Laplacian and graph perimeter are fundamental in the definition of the algorithms. Operators and functionals in the cloud are defined in close resemblance to operators and functionals in the continuum and it is often the case that this resemblance is the starting point for establishing consistency results.