  The task of conditional generation is one of the most important applications
of generative models, and numerous methods have been developed to date based on
the celebrated flow-based models. However, many flow-based models in use today
are not built to allow one to introduce an explicit inductive bias to how the
conditional distribution to be generated changes with respect to conditions.
This can result in unexpected behavior in the task of style transfer, for
example. In this research, we introduce extended flow matching (EFM), a direct
extension of flow matching that learns a "matrix field" corresponding to the
continuous map from the space of conditions to the space of distributions. We
show that we can introduce inductive bias to the conditional generation through
the matrix field and demonstrate this fact with MMOT-EFM, a version of EFM that
aims to minimize the Dirichlet energy or the sensitivity of the distribution
with respect to conditions. We will present our theory along with experimental
results that support the competitiveness of EFM in conditional generation.
