Dimension reduction techniques typically seek an embedding of a high-dimensional point cloud into a low-dimensional Euclidean space which optimally preserves the geometry of the input data. Based on expert knowledge, one may instead wish to embed the data into some other manifold or metric space in order to better reflect the geometry or topology of the point cloud. We propose a general method for manifold-valued multidimensional scaling based on concepts from optimal transport. In particular, we establish theoretical connections between the recently introduced semi-relaxed Gromov-Wasserstein (srGW) framework and multidimensional scaling by solving the Monge problem in this setting. We also derive novel connections between srGW distance and Gromov-Hausdorff distance. We apply our computational framework to analyze ensembles of political redistricting plans for states with two Congressional districts, achieving an effective visualization of the ensemble as a distribution on a circle which can be used to characterize typical neutral plans, and to flag outliers.

Abstract We prove an equivalence between open questions about the embeddability of the space of persistence diagrams and the space of probability distributions (i.e. Wasserstein space). It is known that for many natural metrics, no coarse embedding of either of these two spaces into Hilbert space exists. Some cases remain open, however. In particular, whether coarse embeddings exist with respect to the p-Wasserstein distance for $$1\le p\le 2$$ 1 ≤ p ≤ 2 remains an open question for the space of persistence diagrams and for Wasserstein space on the plane. In this paper, we show that embeddability for persistence diagrams is equivalent to embeddability for Wasserstein space on $$\mathbb {R}^2$$ R 2 . When $$p > 1$$ p > 1 , Wasserstein space on $$\mathbb {R}^2$$ R 2 is snowflake universal (an obstruction to embeddability into any Banach space of non-trivial type) if and only if the space of persistence diagrams is snowflake universal.

In our technology-dependent modern world, it is imperative to monitor the Sun for space weather threats to critical infrastructure. Topological data analysis (TDA) is a new set of mathematical techniques used in data analysis and machine learning. We demonstrate that TDA can robustly detect and classify solar surface and coronal activity. This technique is a promising step toward future application in predictive space weather modeling.

The efficacy of topological data analysis (TDA) has been demonstrated in many different machine learning pipelines, particularly those in which structural characteristics of data are highly relevant. However, TDA's usability in large scale machine learning applications is hindered by the significant computational cost of generating persistence diagrams. In this work, a method that allows this computationally expensive process to be approximated by deep neural networks is proposed. Moreover, the method's practicality in estimating 0-dimensional persistence diagrams across a diverse range of images is shown.

Motivated by the problem of redistricting, we study area-preserving reconfigurations of connected subdivisions of a simple polygon. A connected subdivision of a polygon $\mathcal{R}$, called a district map, is a set of interior disjoint connected polygons called districts whose union equals $\mathcal{R}$. We consider the recombination as the reconfiguration move which takes a subdivision and produces another by merging two adjacent districts, and by splitting them into two connected polygons of the same area as the original districts. The complexity of a map is the number of vertices in the boundaries of its districts. Given two maps with $k$ districts, with complexity $O(n)$, and a perfect matching between districts of the same area in the two maps, we show constructively that $(\log n)^{O(\log k)}$ recombination moves are sufficient to reconfigure one into the other. We also show that $Ω(\log n)$ recombination moves are sometimes necessary even when $k=3$, thus providing a tight bound when $k=O(1)$.