The geometry of touching: optimal transport to cluster playing styles and match outcomes in soccer

Touching distributions, represented as spatial maps of where players interact with the ball during a match, play a fundamental role in shaping team strategy and influencing match outcomes in soccer and other sports. In this work, we propose a novel approach based on optimal transport theory to quantify and compare the spatial distributions of players' touches across matches. By leveraging both balanced and unbalanced optimal transport formulations, we capture not only the spatial structure of touching patterns but also their volumetric differences. Our analysis shows that touching distributions effectively discriminate between match outcomes, with teams exhibiting more consistent patterns in victories than in draws or defeats. Furthermore, we cluster team touching distributions using Wasserstein barycentres and identify dominant playing styles that, while largely stable within outcomes, exhibit notable variations between wins, draws, and losses. Finally, we evaluate the predictive accuracy of the clustering method, demonstrating that spatial touching distributions, reflecting distinct on-field playing styles, have predictive power in discriminating match outcomes.