  Increasingly, statisticians are faced with the task of analyzing complex data
that are non-Euclidean and specifically do not lie in a vector space. To
address the need for statistical methods for such data, we introduce the
concept of Fr\'echet regression. This is a general approach to regression when
responses are complex random objects in a metric space and predictors are in
$\mathcal{R}^p$, achieved by extending the classical concept of a Fr\'echet
mean to the notion of a conditional Fr\'echet mean. We develop generalized
versions of both global least squares regression and local weighted least
squares smoothing. The target quantities are appropriately defined population
versions of global and local regression for response objects in a metric space.
We derive asymptotic rates of convergence for the corresponding fitted
regressions using observed data to the population targets under suitable
regularity conditions by applying empirical process methods. For the special
case of random objects that reside in a Hilbert space, such as regression
models with vector predictors and functional data as responses, we obtain a
limit distribution. The proposed methods have broad applicability. Illustrative
examples include responses that consist of probability distributions and
correlation matrices, and we demonstrate both global and local Fr\'echet
regression for demographic and brain imaging data. Local Fr\'echet regression
is also illustrated via a simulation with response data which lie on the
sphere.
