Optimal transport theory has recently found many applications in machine learning thanks to its capacity to meaningfully compare various machine learning objects that are viewed as distributions. The Kantorovitch formulation, leading to the Wasserstein distance, focuses on the features of the elements of the objects, but treats them independently, whereas the Gromov–Wasserstein distance focuses on the relations between the elements, depicting the structure of the object, yet discarding its features. In this paper, we study the Fused Gromov-Wasserstein distance that extends the Wasserstein and Gromov–Wasserstein distances in order to encode simultaneously both the feature and structure information. We provide the mathematical framework for this distance in the continuous setting, prove its metric and interpolation properties, and provide a concentration result for the convergence of finite samples. We also illustrate and interpret its use in various applications, where structured objects are involved.

In the context of optimal transport (OT) methods, the subspace detour approach was recently proposed by Muzellec and Cuturi. It consists of first finding an optimal plan between the measures projected on a wisely chosen subspace and then completing it in a nearly optimal transport plan on the whole space. The contribution of this paper is to extend this category of methods to the Gromov–Wasserstein problem, which is a particular type of OT distance involving the specific geometry of each distribution. After deriving the associated formalism and properties, we give an experimental illustration on a shape matching problem. We also discuss a specific cost for which we can show connections with the Knothe–Rosenblatt rearrangement.

In this work we address the problem of comparing time series while taking into account both feature space transformation and temporal variability. The proposed framework combines a latent global transformation of the feature space with the widely used Dynamic Time Warping (DTW). The latent global transformation captures the feature invariance while the DTW (or its smooth counterpart soft-DTW) deals with the temporal shifts. We cast the problem as a joint optimization over the global transformation and the temporal alignments. The versatility of our framework allows for several variants depending on the invariance class at stake. Among our contributions we define a differentiable loss for time series and present two algorithms for the computation of time series barycenters under our new geometry. We illustrate the interest of our approach on both simulated and real world data.