Grunnleggende konsepter
The paper introduces two Gromov-Wasserstein-type distances, MGW2 and EW2, to efficiently compare Gaussian mixture models (GMMs) across different dimensions and invariant to isometries.
Sammendrag
The paper introduces two Gromov-Wasserstein-type distances to compare Gaussian mixture models (GMMs):
Mixture Gromov-Wasserstein (MGW2):
MGW2 is a natural "Gromovization" of the Mixture-Wasserstein (MW2) distance, which compares GMMs by restricting the set of admissible transportation couplings to be GMMs themselves.
MGW2 defines a pseudometric on the set of all finite GMMs that is invariant to isometries, but does not directly provide an optimal transportation plan between the points.
Embedded Wasserstein (EW2):
EW2 is shown to be equivalent to the invariant OT distance introduced by Alvarez-Melis et al. (2019), which explicitly encodes the isometric transformation applied to one of the measures.
EW2 can be used as an alternative to Gromov-Wasserstein and allows to derive an optimal assignment between points, but is computationally more expensive than MGW2.
The paper also discusses the practical use of MGW2 on discrete data distributions and the difficulty of designing a transportation plan associated with the MGW2 problem. Finally, the authors illustrate the use of their distances on medium-to-large scale problems such as shape matching and hyperspectral image color transfer.