Generalized Gromov–Wasserstein Transport Models for Heterogeneous Data

• Applicant: Dr. Florian Beier
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 564105454

Gromov—Wasserstein (GW) distances offer a powerful framework in heterogeneous contexts by optimally comparing and aligning the internal geometry of input spaces. In this project, we develop generalized GW models with a focus on both theoretical and practical contributions. In particular, we explore the so-called unbalanced fused GW transport problem which accounts for noisy and labelled data. We want to study conditions under which the induced distance becomes a metric. Furthermore, we analyze the geometry of this metric space. Arising geometric concepts like tangent spaces may yield fruitful practical approximation methods. In addition, we investigate a relaxation of the Wasserstein distance, where the GW distance is used as a regularizer. More precisely, the GW terms are employed to penalize any non-isometric marginal derivations. In practice, the proposed framework is able to jointly embed hetereogeneous data into an arbitrary metric space. Finally, we will we study the GW quantization problem which focuses on seeking approximations of arbitrary spaces in the GW sense with small support size. We are interested in deriving worst-case error bounds of this approximation and producing a methodology on how to compute them. Result may be used for the practical quantization of embedding-free data structures such as graphs and meshes.

• DFG Programme: WBP Fellowship
• International Connection: France
• Hosts: Professor Dr. Stephan Eckstein; Professor Dr. Francois-Xavier Vialard