  Distribution data refers to a data set where each sample is represented as a
probability distribution, a subject area receiving burgeoning interest in the
field of statistics. Although several studies have developed
distribution-to-distribution regression models for univariate variables, the
multivariate scenario remains under-explored due to technical complexities. In
this study, we introduce models for regression from one Gaussian distribution
to another, utilizing the Wasserstein metric. These models are constructed
using the geometry of the Wasserstein space, which enables the transformation
of Gaussian distributions into components of a linear matrix space. Owing to
their linear regression frameworks, our models are intuitively understandable,
and their implementation is simplified because of the optimal transport
problem's analytical solution between Gaussian distributions. We also explore a
generalization of our models to encompass non-Gaussian scenarios. We establish
the convergence rates of in-sample prediction errors for the empirical risk
minimizations in our models. In comparative simulation experiments, our models
demonstrate superior performance over a simpler alternative method that
transforms Gaussian distributions into matrices. We present an application of
our methodology using weather data for illustration purposes.
