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Abstract The Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science and in Machine Learning. With the goal of aiding both the interpretability of dissimilarity measures computed through the Gromov–Wasserstein distance and the assessment of the approximation quality of computational techniques designed to estimate the Gromov–Wasserstein distance, we determine the precise value of a certain variant of the Gromov–Wasserstein distance between unit spheres of different dimensions. Indeed, we consider a two-parameter family$$\{d_{{{\text {GW}}}p,q}\}_{p,q=1}^{\infty }$$ ${\left\{d_{\text{GW}p,q}\right\}}_{p,q=1}^{\infty }$of Gromov–Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameterspandqand the metric of the underlying spaces, we are able to determine the exact value of the distance$$d_{{{\text {GW}}}4,2}$$ $d_{\text{GW}4,2}$between all pairs of unit spheres of different dimensions endowed with their Euclidean distance and their uniform measure. 

Abstract We study a generalization of the classical multidimensional scaling procedure (cMDS) which is applicable in the setting of metric measure spaces. Metric measure spaces can be seen as natural ‘continuous limits’ of finite data sets. Given a metric measure space $${\mathcal{X}} = (X,d_{X},\mu _{X})$$, the generalized cMDS procedure involves studying an operator which may have infinite rank, a possibility which leads to studying its traceability. We establish that several continuous exemplar metric measure spaces such as spheres and tori (both with their respective geodesic metrics) induce traceable cMDS operators, a fact which allows us to obtain the complete characterization of the metrics induced by their resulting cMDS embeddings. To complement this, we also exhibit a metric measure space whose associated cMDS operator is not traceable. Finally, we establish the stability of the generalized cMDS method with respect to the Gromov–Wasserstein distance. 

Abstract We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers$$k\ge 0$$ $k\ge 0$and$$n\ge 1$$ $n\ge 1$we consider the dimensionkVietoris–Rips persistence diagrams ofallsubsets of a given metric space with cardinality at mostn. We call these invariantspersistence setsand denote them as$${\textbf{D}}_{n,k}^{\textrm{VR}}$$ $D_{n,k}^{\text{VR}}$. We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parametersnandk, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which$${\textbf{D}}_{4,1}^{\textrm{VR}}$$ $D_{4,1}^{\text{VR}}$fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a spaceXwith cardinality$$2k+2$$ $2k+2$with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction. 

Abstract One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length$$\ell \ge 0$$ $\ell \ge 0$and the other is the filtration parameter. This new persistence structure, called thepersistent cup module, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter$$\ell $$ $\ell$, we obtain a 1-dimensional persistence module, called the persistent$$\ell $$ $\ell$-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of apersistent invariant, which extends both therank invariant(also referred to aspersistent Betti number), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-definedpersistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called thepersistent LS-category invariant.