Neural Information Processing Systems http://nips.cc/

GW while being several order of magnitudes faster to compute.

Gromov-Wasserstein (GW) distance is a key tool for manifold learning and cross-domain learning, allowing the comparison of distributions that do not live in the same metric space. Because of its high computational complexity, several approximate GW distances have been proposed based on entropy regularization or on slicing, and one-dimensional GW computation. In this paper, we propose a novel approach for comparing two incomparable distributions, that hinges on the idea of distributional slicing, embeddings, and on computing the closed-form Wasserstein distance between the sliced distributions. We provide a theoretical analysis of this new divergence, called distributional sliced embedding (DSE) discrepancy, and we show that it preserves several interesting properties of GW distance including rotation-invariance. We show that the embeddings involved in DSE can be efficiently learned. Finally, we provide a large set of experiments illustrating the behavior of DSE as a divergence in the context of generative modeling and in query framework.

Recently used in various machine learning contexts, the Gromov-Wasserstein distance (GW) allows for comparing distributions that do not necessarily lie in the same metric space. However, this Optimal Transport (OT) distance requires solving a complex non convex quadratic program which is most of the time very costly both in time and memory. Contrary to GW, the Wasserstein distance (W) enjoys several properties (e.g. duality) that permit large scale optimization. Among those, the Sliced Wasserstein (SW) distance exploits the direct solution of W on the line, that only requires sorting discrete samples in 1D. This paper propose a new divergence based on GW akin to SW. We first derive a closed form for GW when dealing with 1D distributions, based on a new result for the related quadratic assignment problem. We then define a novel OT discrepancy that can deal with large scale distributions via a slicing approach and we show how it relates to the GW distance while being $O(n^2)$ to compute. We illustrate the behavior of this so called Sliced Gromov-Wasserstein (SGW) discrepancy in experiments where we demonstrate its ability to tackle similar problems as GW while being several order of magnitudes faster to compute