Generalised Wasserstein Barycentres and the Sliced Wasserstein Distance
The Wasserstein distance defines a metric on the space of positive measures, which allows the definition of Wasserstein Barycentres. We consider a generalisation of this notion of barycentre in which the input measures can reside in different spaces. Within this generalisation, we study the case where the input measures are lower-dimensional projections of a certain measure. The problem of computing a Generalised Wasserstein Barycentre between these projections amounts to reconstructing the initial measure using lower-dimensional information (akin to noiseless tomography). We analyse this Reconstruction Problem, and link the energy it minimises to the Sliced Wasserstein Distance, a variation on the Wasserstein Distance.