分野別セミナー

The Wasserstein and Fisher-Rao distances are two central quantities
arising from the fields of Optimal Transport and Information Geometry,
respectively, together with their applications in machine learning and
statistics. On the set of zero-mean Gaussian densities on Euclidean
space, they both admit closed form formulas. In this talk, we present
their generalization to the infinite-dimensional setting of Gaussian
measures on Hilbert space and Gaussian processes.

1. In general, the exact Fisher-Rao metric formulation is not
generalizable on the set of all Gaussian measures on an
infinite-dimensional Hilbert space. Instead, we show that on the set of
all Gaussian measures which are equivalent to a fixed one, all
finite-dimensional formulas admit direct generalization. By employing
regularization, we then have a formulation that is valid for all
Gaussian measures on Hilbert space.

2. The Wasserstein distance, on the other hand, is valid for all
Gaussian measures on Hilbert space. Nevertheless, we show that by
employing entropic regularization, many favorable theoretical
properties, including convergence and differentiability, can be obtained.

3. In the setting of Gaussian processes, by reproducing kernel Hilbert
space (RKHS) methodology, we obtain consistent finite-dimensional
approximations of the infinite-dimensional quantities that can be
practically employed.