The Gromov–Wasserstein (GW) distance and its fused extension (FGW) are powerful tools for comparing heterogeneous data. Their computation is, however, challenging since both distances are based on non-convex, quadratic optimal transport (OT) problems. Leveraging 1D OT, a sliced version of GW has been proposed to lower the computational burden. Unfortunately, this sliced version is restricted to Euclidean geometry and loses invariance to isometries, strongly limiting its application in practice. To overcome these issues, we propose a novel slicing technique for GW as well as for FGW that is based on an appropriate lower bound, hierarchical OT, and suitable quadrature rules for the underlying 1D OT problems. Our novel sliced FGW significantly reduces the numerical effort while remaining invariant to isometric transformations and allowing the comparison of arbitrary geometries. We show that our new distance actually defines a pseudo-metric for structured spaces that bounds FGW from below and study its interpolation properties between sliced Wasserstein and GW. Since we avoid the underlying quadratic program, our sliced distance is numerically more robust and reliable than the original GW and FGW distance; especially in the context of shape retrieval and graph isomorphism testing.

Facial analysis is a key component in a wide range of applications such as healthcare, autonomous driving, and entertainment. Despite the availability of various facial RGB datasets, the thermal modality, which plays a crucial role in life sciences, medicine, and biometrics, has been largely overlooked. To address this gap, we introduce the T-FAKE dataset, a new large-scale synthetic thermal dataset with sparse and dense landmarks. To facilitate the creation of the dataset, we propose a novel RGB2Thermal loss function, which enables the domain-adaptive transfer of RGB faces to thermal style. By utilizing the Wasserstein distance between thermal and RGB patches and the statistical analysis of clinical temperature distributions on faces, we ensure that the generated thermal images closely resemble real samples. Using RGB2Thermal style transfer based on our RGB2Thermal loss function, we create the large-scale synthetic thermal T-FAKE dataset†with landmark and segmentation annotations. Leveraging our novel T-FAKE dataset, probabilistic landmark prediction, and label adaptation networks, we demonstrate significant improvements in landmark detection methods on thermal images across different landmark conventions. Our models show excellent performance with both sparse 70-point landmarks and dense 478-point landmark annotations. Moreover, our RGB2Thermal loss leads to notable results in terms of perceptual evaluation and temperature prediction.

We propose a new approach for unsupervised alignment of heterogeneous datasets, which maps data from two different domains without any known correspondences to a common metric space. Our method is based on an unbalanced optimal transport problem with Gromov-Wasserstein marginal penalization. It can be seen as a counterpart to the recently introduced joint multidimensional scaling method. We prove that there exists a minimizer of our functional and that for penalization parameters going to infinity, the corresponding sequence of minimizers converges to a minimizer of the so-called embedded Wasserstein distance. Our model can be reformulated as a quadratic, multi-marginal, unbalanced optimal transport problem, for which a bi-convex relaxation admits a numerical solver via block-coordinate descent. We provide numerical examples for joint embeddings in Euclidean as well as non-Euclidean spaces.