A contribution to Optimal Transport on incomparable spaces

TL;DR

This paper develops new Optimal Transport tools, especially Gromov-Wasserstein distance, for comparing and analyzing data in incomparable spaces like graphs and structured data, with applications in machine learning tasks.

Contribution

It introduces novel Optimal Transport methods tailored for incomparable spaces and analyzes their mathematical properties and computational algorithms.

Findings

Gromov-Wasserstein distance effectively compares structured data.

Proposed algorithms enable practical computation of transport in complex spaces.

Applications include classification, data simplification, and domain adaptation.

Abstract

Optimal Transport is a theory that allows to define geometrical notions of distance between probability distributions and to find correspondences, relationships, between sets of points. Many machine learning applications are derived from this theory, at the frontier between mathematics and optimization. This thesis proposes to study the complex scenario in which the different data belong to incomparable spaces. In particular we address the following questions: how to define and apply Optimal Transport between graphs, between structured data? How can it be adapted when the data are varied and not embedded in the same metric space? This thesis proposes a set of Optimal Transport tools for these different cases. An important part is notably devoted to the study of the Gromov-Wasserstein distance whose properties allow to define interesting transport problems on incomparable spaces. More broadly, we analyze the mathematical properties of the various proposed tools, we establish algorithmic solutions to compute them and we study their applicability in numerous machine learning scenarii which cover, in particular, classification, simplification, partitioning of structured data, as well as heterogeneous domain adaptation.