The Gromov–Wasserstein Distance Between Spheres

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 $$\{d_{{{\text {GW}}}p,q}\}_{p,q=1}^{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mrow> <mml:mtext>GW</mml:mtext> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:math> of Gromov–Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameters p and q and the metric of the underlying spaces, we are able to determine the exact value of the distance $$d_{{{\text {GW}}}4,2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>d</mml:mi> <mml:mrow> <mml:mtext>GW</mml:mtext> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> between all pairs of unit spheres of different dimensions endowed with their Euclidean distance and their uniform measure.