Search for: All records

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. 

https://arxiv.org/abs/2301.00246