  The existing Fr\'echet regression is actually defined within a linear
framework, since the weight function in the Fr\'echet objective function is
linearly defined, and the resulting Fr\'echet regression function is identified
to be a linear model when the random object belongs to a Hilbert space. Even
for nonparametric and semiparametric Fr\'echet regressions, which are usually
nonlinear, the existing methods handle them by local linear (or local
polynomial) technique, and the resulting Fr\'echet regressions are (locally)
linear as well. We in this paper introduce a type of nonlinear Fr\'echet
regressions. Such a framework can be utilized to fit the essentially nonlinear
models in a general metric space and uniquely identify the nonlinear structure
in a Hilbert space. Particularly, its generalized linear form can return to the
standard linear Fr\'echet regression through a special choice of the weight
function. Moreover, the generalized linear form possesses methodological and
computational simplicity because the Euclidean variable and the metric space
element are completely separable. The favorable theoretical properties (e.g.
the estimation consistency and presentation theorem) of the nonlinear Fr\'echet
regressions are established systemically. The comprehensive simulation studies
and a human mortality data analysis demonstrate that the new strategy is
significantly better than the competitors.
