  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.
